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Lemma 11.19

$(3 / 40, 31 / 40)$ is an exponent pair. In particular, by Corollary 6.8, $μ (7 / 10) \leq 3 / 40$ .

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Lemma 11.33Preliminary large values theorem

If $1 / 2 \leq σ \leq 1$ and $τ < 8 σ - 5$ , then

LV (σ, τ) \leq max (2 - 2 σ, 6 τ / 5 + (20 - 32 σ) / 5) .

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Lemma 11.29Preliminary large values estimate

If $m \geq 2$ is an integer, $3 / 4 < σ \leq 1$ , and $(k, ℓ)$ is an exponent pair, then

LV (σ, τ) \leq max (2 - 2 σ, m (2 - 4 σ) + m τ, min (X, Y))

where

X := 2 τ / 3 + 4 m (3 - 4 σ) / 3

and

Y := max (τ + 3 m (3 - 4 σ), (k + ℓ) τ / k + k (1 + 2 k + 2 ℓ) (3 - 4 σ) / k) .

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Lemma 5.2Non-asymptotic definition of exponent pair

Let $(k, ℓ)$ be a fixed element of 1. Then the following are equivalent:

$(k, ℓ)$ is an exponent pair.
For every (fixed) $ε > 0$ there exist (fixed) $C, P > 0$ such that, whenever $T \geq N \geq 1$ , $I \subset [N, 2 N]$ , and $F$ is a phase function obeying 3 for for all (fixed) $0 \leq p \leq P$ and $u \in [1, 2]$ , then

$| \sum_{n \in I} e (T F (n / N)) | \leq C (T / N)^{k + ε} N^{ℓ + ε} .$

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Lemma 6.15Growth exponent and zeroes

Let $1 / 2 \leq σ_{0} < 1$ be fixed. Then the assertion $μ (σ_{0}) = 0$ is equivalent to the assertion that for any fixed $ε > 0$ and unbounded $T > 0$ , the number of zeroes $σ + i t$ of the zeta function with $σ \geq σ_{0} + ε$ and $T \leq t \leq T + 1$ is $o (\log T)$ .

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Lemma 10.14Raising to a power, alternative formulation

Let $k$ be a positive integer, $E_{1} \subseteq R^{5}$ be a set satisfying the monotonicity criterion of Lemma 10.13 and

E_{k} := {(σ, τ, ρ, ρ^{*}, s) \in R^{5} : (σ, τ / k, ρ / k, ρ^{*} / k, s / k) \in E_{1}} .

If $E \subseteq E_{1}$ then $E \subseteq E_{k}$ .

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Theorem 10.22

If $(k, ℓ)$ be an exponent pair with $k > 0$ and $(σ, τ, ρ, ρ^{*}, s) \in E$ , then

\begin{aligned} ρ^{*} \leq 1 - 2 σ & + \frac{1}{2} max (ρ + 1, \frac{5}{3} ρ + \frac{τ}{3}, \frac{2 + 3 k + 4 ℓ}{1 + 2 k + 2 ℓ} ρ + \frac{k + ℓ}{1 + 2 k + 2 ℓ} τ) \\ + \frac{1}{2} max (ρ^{*} + 1, 4 ρ, \frac{3}{4} ρ^{*} + ρ + \frac{τ}{2}) . \end{aligned}

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Lemma 10.25Second Guth-Maynard relation

[ 61 , Lemma 1.7 ] If $(σ, τ, ρ, ρ^{*}, s) \in E$ then

ρ^{*} \leq ρ + s - 2 σ .

In particular, from Lemma 10.18 we see for $τ \leq 3 / 2$ that

ρ^{*} \leq max (3 ρ + 1 - 2 σ, 2 ρ + 2 - 2 σ) .

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Corollary 13.5Ingham-Huxley bound

We have $θ_{PNT} \leq \frac{7}{12}$ and $θ_{PNT - AA} \leq \frac{1}{6}$ .

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Corollary 13.6Ingham-Guth-Maynard bound

[ 61 ] We have $θ_{PNT} \leq \frac{17}{30}$ and $θ_{PNT - AA} \leq \frac{2}{15}$ .

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Corollary 13.7

The density hypothesis implies that $θ_{PNT} \leq 1 / 2$ and $θ_{PNT - AA} = 0$ .

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Corollary 13.10

Assuming the Riemann hypothesis, $θ_{gap, 2} = 1$ . (Selberg, 1943 [ 205 ] )
Assuming the Lindelof hypothesis, $θ_{gap, 2} \leq 7 / 6$ . (Heath-Brown, 1979 [ 76 ] )
Unconditionally, $θ_{gap, 2} \leq 23 / 18$ . (Heath-Brown, 1979 [ 77 ] ).

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Lemma 14.13Tong bound for

4 \leq k \leq 11

One has

\begin{aligned} α_{4} & \leq 1 / 2, & α_{5} \leq 4 / 7, & α_{6} \leq 5 / 8, & α_{7} \leq 71 / 107 \\ α_{8} & \leq 41 / 59, & α_{9} \leq 31 / 43, & α_{10} \leq 26 / 35, & α_{11} \leq 19 / 25 \end{aligned}

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Theorem 14.14

[ 80 ] For $4 \leq k \leq 8$ , one has

α_{k} \leq \frac{3 k - 4}{4 k} .

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Theorem 14.15Ivić–Ouellet bound for large

k

[ 111 ] One has

\begin{aligned} α_{10} \leq 27 / 40, α_{11} & \leq 0.6957, α_{12} \leq 0.7130, α_{13} \leq 0.7306, \\ α_{14} \leq 0.7461, α_{15} & \leq 0.75851, α_{16} \leq 0.7691, α_{17} \leq 0.7785, \\ α_{18} \leq 0.7868, α_{19} & \leq 0.7942, α_{20} \leq 0.8009 . \end{aligned}

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Theorem 14.16

[ 110 , Theorem 13.12 ] One can bound $α_{k}$ by

\begin{aligned} (3 k - 4) / 4 k & for 4 \leq k \leq 8 \\ 35 / 54 & for k = 9 \\ 41 / 60 & for k = 10 \\ 7 / 10 & for k = 11 \\ (k - 2) / (k + 2) & for 12 \leq k \leq 25 \\ (k - 1) / (k + 4) & for 26 \leq k \leq 50 \\ (31 k - 98) / 32 k & for 51 \leq k \leq 57 \\ (7 k - 34) / 7 k & for k \geq 58. \end{aligned}

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Theorem 14.18 [ 11 ]

For integer $k \geq 30$ , one has

α_{k} \leq 1 - 1.421 (k - 1.18)^{- 2 / 3} .

Moreover, $α_{k} \leq 1 - 1.889 k^{- 2 / 3}$ for sufficiently large $k$ .

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Theorem 18.4Non-vanishing on the 1-line

One has $ζ (1 + i t) \neq 0$ for any real $t$ .

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Definition 19.1Prime counting function on arithmetic progressions

Suppose $a, q \in Z$ with $gcd (a, q) = 1$ . For each $x \geq 0$ , define

π (x; q, a) = \sum_{\begin{matrix} p \leq x \\ p prime \\ q ∣ (p - a) \end{matrix}} 1.

The ordinary prime counting function can be recovered by $π (x) = π (x; 1, 1)$ .

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Definition 19.2Logarithmic integral function

Define the offset logarithmic integral function for $x \geq 2$ by $Li (x) = \int_{2}^{x} \frac{d u}{\log u}$ . Note that $Li (x) \sim \frac{x}{\log x}$ .

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Theorem 19.5Brun-Titchmarsh theorem (1929) [ 213 ]

If $0 < θ < 1$ and $q \leq x^{θ}$ , then

π (x; q, a) = O (\frac{x}{φ (q) \log x}) + O (x^{6 (1 - θ) / 7}) .

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Theorem 19.6Lint, Richert (1965) [ 220 ]

If $y > q$ , then

π (x + y; q, a) - π (x; q, a) < \frac{2 y}{φ (q) \log (y / q)} min (\frac{3}{2}, 1 + \frac{6}{\log (y / q)}) .

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Theorem 19.7Montgomery, Vaughan (1973) [ 169 ]

If $y > q$ , then

π (x + y; q, a) - π (x; q, a) < \frac{2 y}{φ (q) \log (y / q)} .

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Definition 19.8

θ

and

C_{θ}

Suppose $x > 0$ and $q \in Z$ . Define $θ := \frac{\log q}{\log x}$ , and let $C_{θ} > 0$ be the smallest constant such that

max_{a : gcd (a, q) = 1} π (x; q, a) \leq \frac{(C_{θ} + o (1)) x}{φ (q) \log (x)}

as $x \to \infty$ .

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Definition 19.10Linnik’s constant

L

Define $L$ to be the infimum over all $L^{'} > 0$ where there exists $q_{0} (L^{'}) > 0$ such that for all $q \geq q_{0} (L^{'})$ and $x > q^{L^{'}}$ , $min_{gcd (a, q) = 1} π (x; q, a) > 0$ .

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Theorem 19.11Maynard (2013) [ 164 ]

For sufficiently large $q$ and $x > q^{8}$ , we have

\frac{\log q}{q^{1 / 2}} (\frac{x}{φ (q) \log x}) ≪ π (x; q, a) .

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Theorem 19.12Maynard (2013) [ 164 ]

Let $ϵ > 0$ . There exists $q_{0} (ϵ) > 0$ such that for all $q \geq q_{0} (ϵ)$ ,

\frac{q^{- ϵ} x}{φ (q) \log x} ≪ π (x; q, a) .

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Definition 20.1

Let $A \subset N$ be such that there exists $k$ for which

\underset{k times}{\underset{⏟}{A + A + \dots + A}} = N

Then $A$ is called an additive basis of $N$ . The minimum $k$ for which 1 holds is the order of $A$ .

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Theorem 20.4Lagrange’s Four Square Theorem

We have $g (2) = 4$ ; that is every natural number may be written as the sum of $4$ perfect squares.

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Definition 20.6

Let $G_{1} (k)$ be the smallest number $m$ such that

d (\underset{m times}{\underset{⏟}{A_{k} + \dots + A_{k}}}) = 1

where $d (A)$ represents the natural density of $A$ :

d (A) = lim_{N \to \infty} \frac{# (A \cap J_{N})}{N}

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Lemma 20.8

$G (3) \geq 4$

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Theorem 20.9Linnik [ 151 ]

$G (3) \leq 7$

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Theorem 20.10Kamke

Let $f (x)$ be an integer valued polynomial such that there does not exist $d \in N$ such that $d | f (n)$ for all $n \in N$ . Then for sufficiently large $k$ ,

f (x_{1}) + f (x_{2}) + \dots + f (x_{k}) = m

is solvable for all large enough $m$ .

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Theorem 20.11Wooley

Assuming GRH, then

x_{1}^{2} + x_{2}^{2} + x_{3}^{3} + x_{4}^{3} + x_{5}^{6} + x_{6}^{6} = n

is solvable for sufficiently high $n$ . Furthermore, (4) is not solvable for at most $O ((\log N)^{3 + ϵ})$ integers between $1$ and $N$ .

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Theorem 20.12Liu, Wooley, Yu [ 152 ]

Let $E (N)$ be the number of integers $n \equiv 4 (\mod 24)$ for which (6) is not solvable. For any $ϵ > 0$ ,

E (N) ≪ O (N^{\frac{3}{8} + ϵ})

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Definition 20.13

For $k \in N$ , let $H (k)$ be the minimum integer $s$ such that

p_{1}^{k} + p_{2}^{k} + \dots + p_{s}^{k} = n

is solvable for sufficiently large $n$ whenever $n \equiv s (\mod K (k))$ .

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Theorem 20.14Wooley, Kawada 2001 [ 129 ]

We have

$H (4) \leq 14$
For any positive $A$ ,

$p_{1}^{4} + p_{2}^{4} + \dots + p_{7}^{4} = n$

has at most $O (N (\log N)^{- A})$ exceptions for $n \equiv 7 (\mod 240)$ and $1 \leq n \leq N$ .
$H (k) \leq 21$
For any positive $A$ ,

$p_{1}^{5} + p_{2}^{5} + \dots + p_{11}^{5} = n$

has at most $O (N (\log N)^{- A})$ exceptions for $n$ odd and $1 \leq n \leq N$ .

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Theorem 20.15Kumchev, Wooley 2016 [ 138 ]

For large values of $k$ ,

H (k) \leq (4 k - 2) \log k - (2 \log 2 - 1) k - 3

Table 20.3 summarizes the best bounds on $H (k)$ for $7 \leq k \leq 20$ .

7	8	9	10	11	12	13	14	15	16	17	18	19	20
45	57	69	81	93	107	121	134	149	163	177	193	207	223

Table 20.3 Upper bounds for

H (k)

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Definition 20.16

Define the Schnirelmann density of $A \subset N$ as

σ A = inf_{n \geq 1} \frac{# (A \cap J_{n})}{n}

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Definition 20.17

Define the lower asymptotic density of $A \subset N$ as

δ A = \underset{n \to \infty}{lim inf} \frac{# (A \cap J_{n})}{n}

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Theorem 20.19Schnirelmann [ 204 ]

Let $P$ denote the set of primes. Then, $δ (P + P) > 0$ . Therefore, $P$ is an additive basis for $N$ . The order of $P$ is denoted $C$ and called Schnirelmann’s constant.

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Theorem 20.20Romanoff [ 189 ]

Let $S_{a} = {p + a^{k} : p \in P, k \in N}$ . Then, $σ S_{a} > 0$ for all $a \in N$ . Thus, each integer $n$ may be written as the sum of at most $C_{a}$ primes and $C_{a}$ powers of $a$ , where $C_{a}$ is a constant depending only on $a$ .

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Definition 20.21

$B \subset N$ is called an essential component if $σ (A + B) > σ (A)$ for any $A \subset N$ with $0 < σ A < 1$ .

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Lemma 10.2Basic properties of additive energy

If $W$ is a finite set of reals, then

$E_{1} (W) ≍ \int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq 1} |^{2} d x .$

More generally, for any $r > 0$ we have

$E_{1} (W) ≍ r^{O (1)} \int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq r} |^{2} d x .$
If $W$ is a finite set of reals, then

$E_{1} (W) ≍ \int_{- 1}^{1} | \sum_{t \in W} e (t θ) |^{4} d θ .$
If $W_{1}, \dots, W_{k}$ are finite sets of reals, then

$E_{1} (W_{1} \cup \dots \cup W_{k})^{1 / 4} ≪ E_{1} (W_{1})^{1 / 4} + \dots + E_{1} (W_{k})^{1 / 4} .$
If $W$ is $1$ -separated and contained in an interval of length $T \geq 1$ , then

$(# W)^{2}, (# W)^{4} / T ≪ E_{1} (W) ≪ (# W)^{3} .$
If $W$ is contained in an interval $I$ , which is in turn split into $K$ equally sized subintervals $J_{1}, \dots, J_{K}$ , then

$E_{1} (W)^{1 / 3} ≪ \sum_{k = 1}^{K} E_{1} (W \cap J_{k})^{1 / 3} .$

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Corollary 5.11Additional exponent pairs

The pairs

(\frac{13}{31}, \frac{16}{31}), (\frac{4}{11}, \frac{6}{11}), (\frac{2}{7}, \frac{4}{7}), (\frac{5}{24}, \frac{15}{24}), (\frac{4}{18}, \frac{11}{18})

are all exponent pairs.

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Theorem 20.18Schnirelmann [ 204 ]

Suppose $σ A > 0$ . Then $A$ is an additive basis for $N$ .

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Proposition 2.1Automatic uniformity

Let $E = E_{i}$ be a non-empty variable set, and let $f = f_{i} : E \to C$ be a variable function.

Suppose that $f (x) = O (1)$ for all (variable) $x \in E$ . Then after passing to a subsequence if necessary, the bound is uniform, that is to say, there exists a fixed $C$ such that $| f (x) | \leq C$ for all $x \in E$ .
Suppose that $f (x) = o (1)$ for all (variable) $x \in E$ . Then after passing to a subsequence if necessary, the bound is uniform, that is to say, there exists an infinitesimal $c$ such that $| f (x) | \leq c$ for all $x \in E$ .

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Theorem 14.3Hardy [ 66 ]

One has $β_{2} = 1 / 4$ .

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Theorem 14.4Cramér [ 37 ]

One has $β_{3} = 1 / 3$ .

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Lemma 4.3Non-asymptotic definition of

β

Let $α \geq 0$ and $\overset{―}{β} \in R$ be fixed. Then the following are equivalent:

$β (α) \leq \overset{―}{β}$ .
For every (fixed) $ε > 0$ and $σ > 0$ there exists (fixed) $δ > 0$ , $P \geq 1$ , $C \geq 1$ with the following property: if $T \geq C$ , $T^{α - δ} \leq N \leq T^{α + δ}$ are (fixed) real numbers, $I \subset [N, 2 N]$ is a (fixed) interval, and $F$ is a (fixed) phase function such that

$| F^{(p + 1)} (u) - \frac{d^{p}}{d u^{p}} u^{- σ} | \leq δ$
3

for all (fixed) $0 \leq p \leq P$ and $u \in [1, 2]$ , then

$| \sum_{n \in I} e (T F (n / N)) | \leq C T^{\overset{―}{β} + ε} .$

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Theorem 4.242017 Bourgain bound

One has

β (α) \leq {\begin{cases} \frac{2}{9} + \frac{1}{3} α, & \frac{1}{3} < α \leq \frac{5}{12}, \\ \frac{1}{12} + \frac{2}{3} α, & \frac{5}{12} < α \leq \frac{3}{7}, \\ \frac{13}{84} + \frac{1}{2} α, & \frac{3}{7} < α \leq \frac{1}{2} . \end{cases}

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Definition 4.2Exponent sum growth exponent

For any fixed $α \geq 0$ , let $β (α) \in R$ denote the least possible (fixed) exponent for which the following claim holds: whenever $N, T \geq 1$ are (variable) quantities with $T$ unbounded and $N = T^{α + o (1)}$ , $F$ is a model phase function, and $I \subset [N, 2 N]$ is an interval, then

\sum_{n \in I} e (T F (n / N)) ≪ T^{β (α) + o (1)} .

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Lemma 5.3Duality between exponent pairs and

β

Let $(k, ℓ)$ be in the triangle 1. Then the following are equivalent:

$(k, ℓ)$ is an exponent pair.
$β (α) \leq k + (ℓ - k) α$ for all $0 \leq α \leq 1$ .

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Theorem 4.232017 Heath-Brown bound

For any $α > 0$ and any natural number $k \geq 3$ one has

β (α) \leq α + max (\frac{1 - k α}{k (k - 1)}, - \frac{α}{k (k - 1)}, - \frac{2 α}{k (k - 1)} - \frac{2 (1 - k α)}{k^{2} (k - 1)}) .

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Theorem 4.252020 Heath-Brown bound

[ 46 , Theorem 11.2 ] If $α$ is fixed with $1 \leq 4 α - 1 \leq 2$ (i.e., $1 / 2 \leq α \leq 3 / 4$ ), then

β (α) \leq max (α (1 - \frac{4 α - 1}{4 (4 α - 1) + 8}), \frac{8}{9} α) .

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Theorem 4.172001 Huxley–Kolesnik bound

For any $2 / 5 \leq α \leq 1 / 2$ one has

β (α) \leq max (\frac{7}{80} + \frac{79}{120} α, \frac{3}{32} + \frac{103}{160} α, \frac{9}{40} + \frac{13}{40} α) .

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Theorem 4.121991 Huxley–Kolesnik bound

For any $2 / 5 \leq α \leq 1 / 2$ one has

β (α) \leq max (\frac{1 + 8 α}{22}, \frac{11 + 112 α}{158}, \frac{1 + 17 α}{22}) .

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Theorem 4.131993 Huxley bound

For any $0 \leq α \leq 49 / 114$ , one has

β (α) \leq max (\frac{13}{60} + \frac{7}{20} α, \frac{11}{120} + \frac{13}{20} α) .

Furthermore, for any $49 / 114 \leq α \leq 1 / 2$ , one has

β (α) \leq \frac{89}{570} + \frac{1}{2} α .

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Theorem 4.14Second 1993 Huxley bound

If $0 \leq α \leq 1$ , then $β (α)$ is bounded by

\begin{aligned} \frac{1}{146} (13 + 94 α) & for α \leq \frac{87}{275} \\ \frac{1}{244} (11 + 191 α) & for \frac{87}{275} \leq α \leq \frac{423}{1295} \\ \frac{1}{1282} (89 + 908 α) & for \frac{423}{1295} \leq α \leq \frac{227}{601} \\ \frac{1}{280} (29 + 173 α) & for \frac{227}{601} \leq α \leq \frac{12}{31} \\ \frac{1}{128} (4 + 103 α) & for \frac{12}{31} \leq α \leq 1. \end{aligned}

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Theorem 4.20Huxley bound

For any $1 / 3 \leq α \leq 1 / 2$ , one has

β (α) \leq max (\frac{37 + 59 α}{170}, \frac{63 + 449 α}{690}) .

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Theorem 4.212016 Robert bound

For any $0 < α \leq 3 / 7$ one has

β (α) \leq max (α + \frac{1 - 4 α}{12}, \frac{11}{12} α) .

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Theorem 4.22Second 2016 Robert bound

If $k \geq 4$ and $α \geq - (1 - k α) \frac{k - 1}{2 k - 3}$ then

β (α) \leq α + max (\frac{1 - k α}{2 (k - 1) (k - 2)}, - \frac{1}{2 (k - 1) (k - 2)}) .

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Lemma 4.10Reflection

For any $0 < α < 1$ , we have $β (α) - \frac{α}{2} = β (1 - α) - \frac{1 - α}{2}$ , or equivalently $β (1 - α) = \frac{1}{2} - α + β (α)$ .

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Theorem 4.182002 Robert–Sargos bound

For any $α > 0$ one has

β (α) \leq max (α + \frac{1 - 4 α}{13}, - \frac{7 (1 - 4 α)}{13}) .

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Lemma 4.5Upper semicontinuity

$β$ is an upper semicontinuous function.

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Lemma 4.4Trivial bounds on

β

For any fixed $α > 1$ , we have

β (α) = α - 1.

For fixed $0 \leq α \leq 1$ , we have

\frac{α}{2} \leq β (α) \leq α .

In particular

β (0) = 0.

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Proposition 4.7Van der Corput inequality

For any natural number $k \geq 2$ and any $α > 0$ , one has

β (α) \leq max (α + \frac{1 - k α}{2^{k} - 2}, (1 - 2^{2 - k}) α - \frac{1 - α}{2^{k} - 2}) .

Thus for instance when $k = 2$ we have

β (α) \leq max (\frac{1}{2}, \frac{2 α - 1}{2}),

so in particular

β (1) = \frac{1}{2},

by Lemma 4.4, when $k = 3$ one has

β (α) \leq max (\frac{1 + 3 α}{6}, \frac{6 α - 1}{3}),

and when $k = 4$ one has

β (α) \leq max (\frac{10 α + 1}{14}, \frac{29 α - 2}{28}) .

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Theorem 4.111989 Watt bound

For any $3 / 7 \leq α \leq 1 / 2$ , one has

β (α) \leq \frac{89}{560} + \frac{1}{2} α .

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Corollary 8.5

If $τ \geq 0$ is fixed then ${LV}_{ζ} (σ, τ) = - \infty$ whenever $σ > τ β (1 / τ)$ is fixed. For instance, by 8, one has ${LV}_{ζ} (σ, 1) = - \infty$ whenever $σ > 1 / 2$ is fixed.

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Theorem 13.8

[ 158 ] We have $θ_{gap} \leq 259 / 500 = 0.518$ .

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Corollary 10.31Bourgain large values theorem, optimized version

For each row $(ρ_{0}, α_{1}, α_{2}, S)$ of Table 10.1, one has

ρ \leq ρ_{0} (σ, τ), (σ, τ) \in S .

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Corollary 10.30Bourgain large values theorem, simplified version

[ 20 , Lemma 4.60 ] Let the notation be as above, but additionally assume $ρ \leq min (1, 4 - 2 τ)$ . Then

ρ \leq max (α_{2} + 2 - 2 σ, α_{1} + α_{2} / 2 + 2 - 2 σ, - α_{2} + 2 τ + 4 - 8 σ, - 2 α_{1} + τ + 12 - 16 σ, 4 α_{1} + 2 + max (1, 2 τ - 2) - 4 σ) .

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Theorem 11.21Bourgain result on density hypothesis

The density hypothesis holds for $σ > 25 / 32$ .

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Theorem 11.22Improved Bourgain density hypothesis bound

For $17 / 22 \leq σ \leq 4 / 5$ , one has $A (σ) \leq max (\frac{2}{9 σ - 6}, \frac{9}{8 (2 σ - 1)})$ . Thus one has $A (σ) \leq \frac{9}{8 (2 σ - 1)}$ for $38 / 49 \leq σ \leq 4 / 5$ and $A (σ) \leq \frac{2}{9 σ - 6}$ for $17 / 22 \leq σ \leq 38 / 49$ .

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Theorem 10.29Bourgain large values theorem

[ 20 ] Let $1 / 2 < σ < 1$ and $τ > 0$ , and let $ρ := LV (σ, τ)$ . Let $α_{1}, α_{2} \geq 0$ be real numbers. Then either

ρ \leq max (α_{2} + 2 - 2 σ, - α_{2} + 2 τ + 4 - 8 σ, - 2 α_{1} + τ + 12 - 16 σ)

or else there exists $s \geq 0$ such that

\begin{aligned} \frac{1}{2} max (ρ + 2, 2 ρ + 1, 5 ρ / 4 + τ / 2 + 1) + \frac{1}{2} max (s + 2, 2 s + 1, 5 s / 4 + τ / 2 + 1) \geq \\ max (- 2 α_{1} + 2 σ + s + ρ, - α_{1} - α_{2} / 2 + 2 σ + s / 2 + 3 ρ / 2) . \end{aligned}

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Lemma 9.15Large value theorems from mixed moment bounds

[ 19 , Proposition 2 ] Suppose that $M (1 / 2, A, \geq h) \leq 1$ for some $A \geq 4$ and $h \geq 0$ . Then one has

LV (σ, τ) \leq max (α + 2 - 2 σ, - α + τ + A / 2 - 2 A (σ - 1 / 2))

whenever $1 / 2 \leq σ \leq 1$ , $τ > 0$ , and $0 \leq α \leq 1 - σ$ is such that

σ - \frac{1}{2} > \frac{τ h}{2} + \frac{1}{4} .

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Lemma 9.16Zero density theorems from mixed moment bounds

[ 19 , Proposition 5 ] Suppose that $M (1 / 2, 6, \geq h) \leq 1$ for some $h \geq 0$ . Then for any $1 / 2 \leq α < σ < 1$ , one has

A (σ) \leq max (\frac{μ (α)}{σ - α}, \frac{3}{8 σ - 5}, \frac{6 h}{4 σ - 3}) .

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Theorem 11.23Bourgain zero density theorem

[ 19 , Proposition 3 ] Let $(k, ℓ)$ be an exponent pair with $k < 1 / 5$ , $ℓ > 3 / 5$ , and $15 ℓ + 20 k > 13$ . Then, for any $σ > \frac{ℓ + 1}{2 (k + 1)}$ , one has

A (σ) \leq \frac{4 k}{2 (1 + k) σ - 1 - ℓ}

assuming either that $k < \frac{11}{85}$ , or that $\frac{11}{85} < k < \frac{1}{5}$ and $σ > \frac{144 k - 11 ℓ - 11}{170 k - 22}$ .

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Corollary 11.24Special case of Bourgain’s zero density theorem

[ 19 , Corollary 4 ] One has

A (σ) \leq \frac{4}{30 σ - 25}

for $\frac{15}{16} \leq σ \leq 1$ and

A (σ) \leq \frac{2}{7 σ - 5}

for $\frac{17}{19} \leq σ \leq \frac{15}{16}$ .

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Theorem 11.322000 Bourgain zero density theorem

[ 16 ] One has $A (σ) \leq 3 / 2 σ$ for $3734 / 4694 \leq σ \leq 1$ .

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Corollary 11.25Optimized Bourgain zero density bound

One has

A (σ) \leq {\begin{cases} \frac{11}{12 (4 σ - 3)} & \frac{3}{4} < σ \leq \frac{14}{15}, \\ \frac{391}{2493 σ - 2014} & \frac{14}{15} < σ \leq \frac{2841}{3016}, \\ \frac{22232}{163248 σ - 134765} & \frac{2841}{3016} < σ \leq \frac{859}{908}, \\ \frac{356}{2742 σ - 2279} & \frac{859}{908} < σ \leq \frac{1625}{1692}, \\ \frac{2609588}{20732766 σ - 17313767} & \frac{1625}{1692} < σ \leq \frac{3334585}{3447984}, \\ \frac{75872}{9 (81024 σ - 69517)} & \frac{3334585}{3447984} < σ \leq \frac{974605}{1005296}, \\ \frac{288}{3616 σ - 3197} & \frac{974605}{1005296} < σ \leq \frac{5857}{6032}, \\ \frac{86152}{1447460 σ - 1311509} & \frac{5857}{6032} < σ < 1. \end{cases}

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Theorem 20.7Brudern and Wooley 2022 [ 22 ]

For all $k \geq 1$ ,

G (k) < k (\log k + C_{1}) + C_{2}

Furthermore,

G (k) \leq ⌈ k (\log k + 4.20032) ⌉

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Lemma 10.3Cauchy–Schwarz and double

ζ

-sums

[ 16 , Lemma 3.4 ] If $W, W^{'}$ are finite sets of reals, $N > 1$ , and $a_{n}$ is a $1$ -bounded sequence for $n \in [N, 2 N]$ , then

\sum_{t \in W, t^{'} \in W^{'}} {| \sum_{n \in [N, 2 N]} a_{n} n^{- i t} |}^{2} \leq S (N, W)^{1 / 2} S (N, W^{'})^{1 / 2} .

In particular

\sum_{t \in W} {| \sum_{n \in [N, 2 N]} a_{n} n^{- i t} |}^{2} \leq S (N, W) .

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Theorem 11.36Chen-Debruyne-Vidas density theorem

[ 26 ] For any $279 / 314 \leq σ \leq 17 / 18$ , one has $A (σ) \leq \frac{24}{30 σ - 11}$ .

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Lemma 9.17Chen-Debruyne-Vidas large values theorem

[ 26 , Lemma A.1 ] Let $1 / 2 \leq σ \leq 1$ and $τ \geq \frac{30 σ - 11}{8}$ be fixed. Let $q_{0}, A_{0}, q_{1}, A_{1}, h$ be fixed quantities such that $M (1 / 2, q_{0}, \geq h) \leq A_{0}$ and $M (1 / 2, q_{0}, \leq h) \leq A_{1}$ . Suppose that $ρ \leq LV (σ, τ)$ is such that

\frac{24 (1 - σ)}{30 σ - 11} τ \leq ρ \leq 1.

Then for any $α_{1} \geq 0$ and $0 \leq α_{2} \leq τ$ , one has

ρ \leq max (2 - 2 σ + α_{2}, - 2 α_{1} - (A_{0} - 1) α_{2} + A_{0} τ + (3 - 4 σ) q_{0} / 2, - 2 α_{1} + (A_{1} - 1) α_{2} + A_{1} τ + (3 - 4 σ) q_{1} / 2, 8 α_{1} / 7 + 4 α_{2} / 7 + 16 (1 - σ) / 7 + 6 (10 σ - 9) τ / 7 (30 σ - 11), 16 α_{1} / 3 + 4 (3 - 4 σ) / 3 + 2 (10 σ - 9) τ / (30 σ - 11), 5 α_{1} / 3 + α_{2} / 6 + 2 (3 - 4 σ) / 3 + (1 / 3 + (10 σ - 9) / (30 σ - 11)) τ) .

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Theorem 4.26Combined bound

For $X \leq α \leq Y$ , one has $β (α) \leq β_{0} (α)$ , where $β_{0}, X, Y$ are given by Table 4.3.

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Theorem 19.9From exponent pairs to Brun–Titchmarsh

[ 234 , Theorem 1.4 ] If $(k, ℓ)$ is an exponent pair, then

C_{θ} \leq \frac{4}{(3 + k - ℓ) - (3 + 3 k - ℓ) θ}

whenever

\frac{1 + k - ℓ}{2 + 2 k - 2 ℓ} \leq θ \leq \frac{1 + k - ℓ}{1 + 2 k - ℓ} .

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Lemma 14.2

d_{1}

exponent

One has $α_{1} = β_{1} = 0$ .

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Theorem 14.7

[ 146 , Theorem 1.2 ] One has $α_{2} \leq α^{*} = 0.314483 \dots$ , where $α^{*}$ is the solution to the equation

\frac{8}{25} α - \frac{(\sqrt{2 (1 + 14 α)} - 5 \sqrt{- 1 + 8 α})^{2}}{200} + \frac{51}{200} = α

on the interval $α \in [0.3, 0.35]$ .

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Definition 14.1Divisor sum exponents

Let $k \geq 1$ be a fixed integer. Then, $α_{k}$ is the least (fixed) exponent for which

Δ_{k} (x) ≪ x^{α_{k} + o (1)}

for unbounded $x > 0$ . Furthermore, $β_{k}$ is the least (fixed) exponent for which

{(\frac{1}{x} \int_{1}^{x} (Δ_{k} (t))^{2} d t)}^{1 / 2} ≪ x^{β_{k} + o (1)}

for unbounded $x > 0$ (in both definitions, the implied constant may depend on $k$ ).

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Theorem 14.8

[ 133 ] One has $α_{3} \leq 43 / 96$ .

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Lemma 14.5Lower bound on

α_{k}

and

β_{k}

For all $k \geq 1$ , one has

α_{k} \geq β_{k} \geq \frac{1}{2} - \frac{1}{2 k} .

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Lemma 10.17

[ 110 , Lemma 11.2 ] If $(k, ℓ)$ is an exponent pair with $k > 0$ , and $(σ, τ, ρ, ρ^{*}, s) \in E$ , then

s \leq max (ρ + 1, 5 ρ / 3 + τ / 3, \frac{2 + 3 k + 4 ℓ}{1 + 2 k + 2 ℓ} ρ + \frac{k + ℓ}{1 + 2 k + 2 ℓ} τ) + 1.

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Lemma 10.16

If $(σ, τ, ρ, ρ^{*}, s) \in E$ with $τ < 1$ , then $s \leq max (ρ + 1, 2 ρ) + 1$ .

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Definition 10.1Additive energy

Let $W$ be a finite set of real numbers. The additive energy $E_{1} (W)$ of such a set is defined to be the number of quadruples $(t_{1}, t_{2}, t_{3}, t_{4}) \in W$ such that

| t_{1} + t_{2} - t_{3} - t_{4} | \leq 1.

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Lemma 10.8

For any fixed $1 / 2 \leq σ \leq 1, τ \geq 0$ , we have

LV (σ, τ) = sup {ρ : (σ, τ, ρ, ρ^{*}, s) \in E}

and

{LV}_{ζ} (σ, τ) = sup {ρ : (σ, τ, ρ, ρ^{*}, s) \in E_{ζ}}

In particular, we have $ρ \leq LV (σ, τ)$ for all $(σ, τ, ρ, ρ^{*}, s) \in E$ , and $ρ \leq {LV}_{ζ} (σ, τ)$ for all $(σ, τ, ρ, ρ^{*}, s) \in E_{ζ}$ .

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Lemma 10.4Energy controlled by third moment

Suppose that $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is a large value pattern with $T \geq 1$ and $1 \leq N ≪ T^{O (1)}$ . Then

V^{2} E_{1} (W) ≪ T^{o (1)} \sum_{n, m \in [N, 2 N]} | R_{W} (n / m) |^{3} + T^{- 50} .

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Corollary 7.11Converting an exponent pair to a large values theorem

If $(k, ℓ)$ is an exponent pair, and $1 / 2 \leq σ \leq 1$ , and $τ \geq 0$ are fixed, then

LV (σ, τ) \leq max (2 - 2 σ, 2 - 2 σ + τ - \frac{2 σ + k - ℓ - 1}{k}) .

In particular, the Montgomery conjecture holds for $τ \leq \frac{2 σ + k - ℓ - 1}{k}$ .

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Corollary 5.4Exponent pairs closed and convex

The set of exponent pairs is closed and convex.

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Lemma 5.7

The exponent pair conjecture is equivalent to $β (α) = α / 2$ holding true for all $0 \leq α \leq 1$ .

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Definition 5.1Exponent pair

An exponent pair is a (fixed) element $(k, ℓ)$ of the triangle

{(k, ℓ) \in R^{2} : 0 \leq k \leq 1 / 2 \leq ℓ \leq 1, k + ℓ \leq 1}

with the following property: for all model phase functions $F$ , all $T \geq N \leq 1$ , and all intervals $I \subset [N, 2 N]$ , one has

\sum_{n \in I} e (T F (n / N)) ≪ (T / N)^{k + o (1)} N^{ℓ + o (1)}

whenever $T \geq N \geq 1$ , $I$ is an interval in $[N, 2 N]$ , and $F \in U$ .

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Corollary 6.8Exponent pairs and

μ

If $(k, ℓ)$ is an exponent pair, then

μ (ℓ - k) \leq k .

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Proposition 5.5Trivial exponent pairs

$(0, 1)$ and $(1 / 2, 1 / 2)$ are exponent pairs.

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Lemma 6.10

The exponent pair conjecture implies the Lindelöf hypothesis.

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Theorem 5.13Exponent pairs from the Bombieri–Iwaniec method

The following pairs are exponent pairs:

$(\frac{2}{13}, \frac{35}{52})$ [ 98 ] ;
$(\frac{6299}{43860}, \frac{29507}{43860})$ [ 92 , Table 17.3 ] ;
$(\frac{771}{8116}, \frac{1499}{2029})$ [ 202 , p. 285 ] ;
$(\frac{21}{232}, \frac{173}{232})$ [ 202 , p. 286 ] ;
$(\frac{1959}{21656}, \frac{16135}{21656})$ [ 202 , p. 286 ] ;
$(\frac{516247}{6629696}, \frac{5080955}{6629696})$ [ 102 ] , [ 92 , Table 19.2 ] , [ 193 ] .

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Theorem 5.14Exponent pairs from derivative tests

$(k, 1 - m k)$ is an exponent pair when

$k = \frac{1}{13}$ and $m = 3$ [ 193 , Theorem 1 ] ;
$k = \frac{1}{204}$ and $m = 7$ [ 203 , p. 231 ] ;
$k = \frac{1}{130}$ and $m = 8$ [ 192 , (1.1) ] ;
$k = \frac{7}{2640}$ and $m = 8$ [ 203 , p. 231 ] ;
$k = \frac{1}{716}$ and $m = 9$ [ 203 , p. 231 ] ;
$k = \frac{1}{649}$ and $m = 9$ [ 198 ] ;
$k = \frac{7}{4540}$ and $m = 9$ [ 192 , (1.2) ] ;
$k = \frac{1}{615}$ and $m = 9$ [ 192 , (1.1) ] ;
$k = \frac{1}{915}$ and $m = 10$ [ 195 , Th é or è me 2 ] .

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Theorem 5.23More new exponent pairs

The following are exponent pairs:

(\frac{311}{4822}, \frac{3799}{4822}), (\frac{80219}{1298878}, \frac{515638}{649439}) .

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Lemma 15.3

If $(k, ℓ)$ is an exponent pair, and RH holds, then

θ_{Pythag} \leq max (\frac{1}{3} - \frac{5}{6} \frac{k + ℓ - 3 / 2}{4 (k + ℓ) - 7}, \frac{1}{2} - \frac{3}{2} \frac{k + ℓ - 3 / 2}{4 (k + ℓ) - 7})

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Corollary 11.311980-1984 Ivic zero density bound

[ 109 ] , [ 110 , Theorem 11.4 ] One can bound $A (σ$ ) by

\begin{aligned} \frac{3}{2 σ} & for \frac{3831}{4791} \leq σ \leq 1; \\ \frac{9}{7 σ - 1} & for \frac{41}{53} \leq σ \leq 1; \\ \frac{6}{5 σ - 1} & for \frac{13}{17} \leq σ \leq 1; \\ \frac{15}{13 σ - 3} & for \frac{127}{167} \leq σ \leq 1; \\ \frac{9}{8 σ - 2} & for \frac{47}{62} \leq σ \leq 1; \end{aligned}

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Definition 20.2

For any $k \geq 1$ let $A_{k} = {n^{k} : n \in N \cup {0}}$ . Let $g (k)$ be the order of $A_{k}$ when it exists. That is, $g (k)$ is the minimum number of $k$ powers needed to write any natural number as the sum of (not necessarily unique) $g (k)$ many $k$ powers including 0.

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Definition 20.3

For any $k \geq 1$ , let $G (k)$ be the minimum $m$ such that there exists $N \geq 1$ for which

\underset{m times}{\underset{⏟}{A_{k} + \dots + A_{k}}} = N ∖ J_{N} .

where $J_{N} = {1, \dots, N}$ . That is, $G (k)$ is the minimum number of $k$ powers such that every sufficiently large integer may be written as the sum of (not necessarily unique) $G (k)$ many $k$ powers including 0.

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Proposition 13.9

We have

θ_{gap, 2} \leq max (2 - \frac{2}{‖ A ‖_{\infty}}, sup_{1 / 2 \leq σ \leq 1} max (α (σ), β (σ)))

where

α (σ) := 4 σ - 2 + 2 \frac{B (σ) (1 - σ) - 1}{B (σ) - A (σ)}

and

β (σ) := 4 σ - 2 + \frac{B (σ) (1 - σ) - 1}{A (σ)}

where $A (σ), B (σ)$ are any upper bounds for $A (σ), A^{*} (σ)$ respectively.

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Lemma 10.24Guth-Maynard relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ then

ρ \leq max (2 - 2 σ, 1 - 2 σ + max (S_{1}, S_{2}, S_{3}) / 3)

where $S_{1}, S_{2}, S_{3}$ are real numbers with

S_{1} \leq - 10,

S_{2} \leq max (2 + 2 ρ, τ + 1 + (2 - 1 / k) ρ, 2 + 2 ρ + (τ / 2 - 3 ρ / 4) / k)

for any positive integer $k$ and

S_{3} \leq 2 τ + ρ / 2 + ρ^{*} / 2

and also

S_{3} \leq max (2 τ + 3 ρ / 2, τ + 1 + ρ / 2 + ρ^{*} / 2) .

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Lemma 10.26Third Guth-Maynard relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ and $1 \leq τ \leq 4 / 3$ , then

ρ^{*} \leq max (ρ + 4 - 4 σ, 21 ρ / 8 + τ / 4 + 1 - 2 σ, 3 ρ + 1 - 2 σ) .

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Theorem 11.16Guth–Maynard bound

For any $1 / 2 < σ < 1$ , one has $A (σ) \leq \frac{15}{3 + 5 σ}$ .

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Theorem 10.28Additional Guth–Maynard large values estimate

For any $1 / 2 < σ < 1$ , $τ \geq 1$ , and natural number $k$ one has

\begin{aligned} LV (σ, τ) \leq max ( & 2 - 2 σ, 5 - 6 σ, (4 - 6 σ) \frac{k}{k + 1} + \frac{k}{k + 1} τ, (5 - 6 σ) \frac{4 k}{4 k + 3} + \frac{2}{4 k + 3} τ, \\ 2 - 4 σ + \frac{4}{3} τ, 3 - 4 σ + \frac{τ}{2}, \frac{9}{2} - 7 σ + τ, \frac{72 - 112 σ}{19} + \frac{18}{19} τ) . \end{aligned}

If one specializes to the case $σ \geq 7 / 10$ and $1 \leq τ \leq 6 / 5$ , one then has

LV (σ, τ) \leq max (2 - 2 σ, 3 - 4 σ + \frac{τ}{2}, (4 - 6 σ) \frac{k}{k + 1} + \frac{k}{k + 1} τ, (5 - 6 σ) \frac{4 k}{4 k + 3} + \frac{2}{4 k + 3} τ) .

and also

LV (σ, τ) \leq max (2 - 2 σ, 3 - 4 σ + \frac{τ}{2}, \frac{46 - 60 σ}{5} + \frac{30 σ - 21}{5} τ) .

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Theorem 10.27Guth–Maynard large values theorem

[ 61 , Theorem 1.1 ] One has

LV (σ, τ) \leq max (2 - 2 σ, 18 / 5 - 4 σ, τ + 12 / 5 - 4 σ) .

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Lemma 11.30General zero density estimate

[ 110 , (11.76), (11.77) ] If $(k, ℓ)$ is an exponent pair, and $m \geq 2$ an integer, then

A (σ) \leq \frac{3 m}{(3 m - 2) σ + 2 - m}

whenever

\begin{aligned} σ & \geq min (\frac{6 m^{2} - 5 m + 2}{8 m^{2} - 7 m + 2}, \\ max (\frac{9 m^{2} - 4 m + 2}{12 m^{2} - 6 m + 2}, \frac{3 m^{2} (1 + 2 k + 2 ℓ) - (4 k + 2 ℓ) m + 2 k + 2 ℓ}{4 m^{2} (1 + 2 k + 2 ℓ) - (6 k + 4 ℓ) m + 2 k + 2 ℓ})) . \end{aligned}

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Corollary 5.21Set of exponent pairs

[ 218 , Theorem 1.3 ] Let $H$ be the convex hull $(0, 1)$ , $(1 / 2, 1 / 2)$ , and of $(k_{n}, ℓ_{n})$ for $n \in Z$ , where $(k_{0}, ℓ_{0}) := 13 / 84$ , $(k_{n}, ℓ_{n})$ for $n = 1, 2, 3, 4$ is defined by Theorem 5.20, $(k_{n}, ℓ_{n}) := A (k_{n - 4}, ℓ_{n - 4})$ for $5 \leq n \leq 8$ , $(k_{n}, ℓ_{n}) := (p_{n}, q_{n})$ for $n > 9$ (with $(p_{n}, q_{n})$ defined by Theorem 5.17), and $(k_{- n}, ℓ_{- n}) := B (k_{n}, ℓ_{n})$ for $n \geq 0$ . Then all elements of $H$ are exponent pairs.

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Theorem 9.7Heath-Brown twelfth moment estimate

[ 75 ] $M (1 / 2, 12) \leq 2$ . Equivalently (by Lemma 9.5), one has ${LV}_{ζ} (σ, τ) \leq 2 τ - 12 (σ - 1 / 2)$ for all $τ \geq 2$ and $1 / 2 \leq σ \leq 1$ .

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Theorem 9.8Auxiliary Heath-Brown estimate

For $τ \geq 2$ and $1 / 2 \leq σ \leq 1$ , one has

{LV}_{ζ} (σ, τ) \leq min (τ - 6 (σ - 1 / 2), 5 τ - 32 (σ - 1 / 2)) .

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Lemma 14.17Heath-Brown bound for large

k

For any $k \geq 2$ , one has

α_{k} \leq 1 - 0.849 k^{- 2 / 3} .

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Theorem 11.18Heath-Brown zero density theorem

[ 79 ] For $11 / 14 \leq σ < 1$ , one has $A (σ) \leq \frac{9}{7 σ - 1}$ (in particular, this recovers Theorem 11.17). For any $3 / 4 \leq σ \leq 1$ , one has $A (σ) \leq max (\frac{3}{10 σ - 7}, \frac{4}{4 σ - 1})$ (which is a superior bound when $σ \geq 20 / 23$ ).

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Theorem 11.20Improved Heath-Brown zero density theorem

For any $7 / 10 < σ \leq 1$ , one has $A (σ) \leq \frac{3}{10 σ - 7}$ .

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Lemma 10.18Heath-Brown bound on double sums

If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then

s \leq max (ρ + 1, 2 ρ, 5 ρ / 4 + τ / 2) + 1.

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Theorem 12.6Heath-Brown’s additive energy bound

[ 79 , Theorem 2 ] Let $1 / 2 \leq σ \leq 1$ be fixed. Then one can bound $A^{*} (σ)$ by

\begin{aligned} \frac{10 - 11 σ}{(2 - σ) (1 - σ)} & for 1 / 2 \leq σ \leq 2 / 3; \\ \frac{18 - 19 σ}{(4 - 2 σ) (1 - σ)} & for 2 / 3 \leq σ \leq 3 / 4; \\ \frac{12}{4 σ - 1} & for 3 / 4 \leq σ \leq 1. \end{aligned}

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Corollary 10.21Simplified Heath-Brown relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ and $τ \leq 3 / 2$ , then

ρ^{*} \leq max (3 ρ + 1 - 2 σ, ρ + 4 - 4 σ, 5 ρ / 2 + (3 - 4 σ) / 2) .

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Lemma 7.15Second Heath-Brown large values theorem

If $3 / 4 < σ \leq 1$ and $τ \geq 0$ are fixed, then

LV (σ, τ) \leq max (2 - 2 σ, k τ + k (2 - 4 σ), 2 τ / 3 + k (12 - 16 σ) / 3)

for any positive integer $k$ .

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Theorem 7.14Heath-Brown large values theorem, optimized

Let $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ be fixed. One has

LV (σ, τ) \leq max (2 - 2 σ, 10 + τ - 13 σ) .

In particular, the Montgomery conjecture holds for $τ \leq 11 σ - 8$ .

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Theorem 6.14Heath-Brown [ 83 ]

μ

bounds

For any integer $k \geq 3$ , one has

μ (1 - \frac{3 k^{2} - 3 k + 2}{k (k - 1)^{2} (k + 2)}) \leq \frac{2}{(k - 1)^{2} (k + 2)} .

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Theorem 10.20Heath-Brown relation

[ 79 , (33) ] If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then one has

ρ^{*} \leq 1 - 2 σ + \frac{1}{2} max (ρ + 1, 2 ρ, \frac{5}{4} ρ + \frac{τ}{2}) + \frac{1}{2} max (ρ^{*} + 1, 4 ρ, \frac{3}{4} ρ^{*} + ρ + \frac{τ}{2}) .

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Lemma 10.23Second Heath-Brown relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ then

ρ \leq max (2 - 2 σ, ρ^{*} / 4 + max (τ / 4 + k (3 - 4 σ) / 4, k τ / 4 + k (1 - 2 σ) / 2))

for any positive integer $k$ .

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Theorem 5.161996 Heath–Brown sequence

[ 217 , (6.17.4) ] For any integer $m \geq 3$ , the pair

(\frac{1}{25 m^{2} (m - 2) \log m}, 1 - \frac{1}{25 m^{2} \log m})

is an exponent pair.

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Theorem 5.172017 Heath–Brown sequence

[ 83 , Theorem 2 ] For any integer $m \geq 3$ , the pair

(p_{m}, q_{m}) := (\frac{2}{(m - 1)^{2} (m + 2)}, 1 - \frac{3 m - 2}{m (m - 1) (m + 2)})

is an exponent pair.

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Theorem 7.13Heath-Brown large values theorem, preliminary form

Let $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ be fixed. If $LV (σ, τ) \leq ρ$ then

LV (σ, τ) \leq max (2 - 2 σ, \frac{11}{12} ρ + \frac{3}{2} + \frac{τ}{6} - 2 σ)

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Theorem 20.5Linnik [ 147 ]

$g (k)$ exists for all $k \geq 1$ .

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Lemma 14.12Hardy–Littlewood bound for

k \geq 4

For $k \geq 4$ , one has

α_{k} \leq 1 - \frac{3}{k + 2} .

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Lemma 8.12Halász–Montgomery inequality

For any $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ , we have

LV (σ, τ) \leq max (2 - 2 σ, 1 - 2 σ + sup_{\overset{1 \leq τ^{'} \leq τ}{max (1 / 2, 2 σ - 1) \leq σ^{'} \leq 1}} σ^{'} + {LV}_{ζ} (σ^{'}, τ^{'})) .

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Theorem 8.14Halász-Turán large values theorem

[ 63 , Theorem 1 ] On the Lindelöf hypothesis, one has the Montgomery conjecture whenever $σ > 3 / 4$ .

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Theorem 11.15Huxley bound

[ 89 ] For any $1 / 2 < σ < 1$ , one has $A (σ) \leq \frac{3}{3 σ - 1}$ . (In particular, the density hypothesis holds for $σ \geq 5 / 6$ .)

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Theorem 7.12Huxley large values theorem

[ 89 , Equation (2.9) ] Let $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ be fixed. Then one has

LV (σ, τ) \leq max (2 - 2 σ, 4 + τ - 6 σ) .

In particular, one has the Montgomery conjecture for $τ \leq 4 σ - 2$ .

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Theorem 4.161996 Huxley table

One can bound $β (α)$ by $β_{0} (α)$ for $X \leq α \leq Y$ for $β_{0}, X, Y$ given by Tables 4.1, 4.2.

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Theorem 5.15Huxley sequence

[ 92 , Table 17.3 ] For any integer $m \geq 1$ , the pair

(\frac{169}{1424 \times 2^{m} - 338}, 1 - \frac{169}{1424 \times 2^{m} - 338} \frac{712 m + 1577}{712})

is an exponent pair.

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Theorem 12.8

For $7 / 10 \leq σ \leq 3 / 4$ , one has

A^{*} (σ) \leq max (\frac{5 (18 - 19 σ)}{2 (5 σ + 3) (1 - σ)}, \frac{2 (45 - 44 σ)}{(2 σ + 15) (1 - σ)}) .

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Theorem 12.9

For $3 / 4 \leq σ \leq 4 / 5$ , one has

A^{*} (σ) \leq max (\frac{197 - 220 σ}{8 (5 σ - 1) (1 - σ)}, \frac{3 (29 - 30 σ)}{5 (5 σ - 1) (1 - σ)}, \frac{4 (10 - 9 σ)}{5 (4 σ - 1) (1 - σ)})

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Theorem 12.10

For $664 / 877 \leq σ \leq 31 / 40$ , one has

A^{*} (σ) \leq max (\frac{72 - 91 σ}{7 (11 σ - 8) (1 - σ)}, \frac{5 (18 - 19 σ)}{2 (5 σ + 3) (1 - σ)}) .

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Theorem 12.11

For $42 / 55 \leq σ \leq 79 / 103$ , one has

A^{*} (σ) \leq max (\frac{18 - 19 σ}{6 (15 σ - 11) (1 - σ)}, \frac{3 (18 - 19 σ)}{4 (4 σ - 1) (1 - σ)}) .

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Theorem 12.12

For $79 / 103 \leq σ \leq 84 / 109$ , one has

A^{*} (σ) \leq max (\frac{18 - 19 σ}{2 (37 σ - 27) (1 - σ)}, \frac{5 (18 - 19 σ)}{2 (13 σ - 3) (1 - σ)}) .

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Theorem 12.13

For $84 / 109 \leq σ \leq 5 / 6$ , one has

A^{*} (σ) \leq max (\frac{18 - 19 σ}{9 (3 σ - 2) (1 - σ)}, \frac{4 (10 - 9 σ)}{5 (4 σ - 1) (1 - σ)}) .

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Theorem 12.7

For $3 / 4 \leq σ \leq 5 / 6$ one has

A^{*} (σ) \leq max (\frac{18 - 19 σ}{2 (3 σ - 1) (1 - σ)}, \frac{4 (10 - 9 σ)}{5 (4 σ - 1) (1 - σ)}) .

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Lemma 4.9

In Definition 4.2, one can take the interval $I$ to be $[N, 2 N]$ .

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Corollary 9.14Specific mixed moments

[ 110 , (8.56) ] $M (1 / 2, 6, \geq 11 / 72) \leq 1$ and $M (1 / 2, 24, \leq 11 / 72) \leq 15 / 4$ .

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Lemma 8.16Second Ivic large values theorem

[ 110 , (11.40) ] For any $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ , one has

LV (σ, τ) \leq max (2 - 2 σ, τ + 9 - 12 σ, 3 τ + 19 (3 - 4 σ) / 2) .

In particular, optimizing using subdivision (Lemma 7.7) we have

LV (σ, τ) \leq max (2 - 2 σ, τ + 9 - 12 σ, τ - \frac{84 σ - 65}{6}) .

This implies the Montgomery conjecture for

τ \leq min (10 σ - 7, 12 σ - \frac{53}{6}) .

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Theorem 8.15First Ivic large values theorem

[ 110 , Lemma 8.2 ] If $τ \geq 0$ and $1 / 2 < σ < σ^{'} < 1$ are fixed, then

LV (σ^{'}, τ) \leq max (2 - 2 σ^{'}, τ - f (σ) (σ^{'} - σ))

where $f (σ)$ is equal to

\begin{aligned} \frac{2}{3 - 4 σ} & for 1 / 2 < σ \leq 2 / 3; \\ \frac{10}{7 - 8 σ} & for 2 / 3 \leq σ \leq 11 / 14; \\ \frac{34}{15 - 16 σ} & for 11 / 14 \leq σ \leq 13 / 15; \\ \frac{98}{31 - 32 σ} & for 13 / 15 \leq σ \leq 57 / 62; \\ \frac{5}{1 - σ} & for 57 / 62 \leq σ < 1. \end{aligned}

In particular, the Montgomery conjecture holds for this choice of $σ^{'}$ if

τ \leq sup_{1 / 2 < σ < σ^{'}} f (σ) (σ^{'} - σ) + 2 - 2 σ^{'} .

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Lemma 9.9Ivic’s table of moment bounds

[ 110 , Theorem 8.4 ] We have $M (σ, A) = 1$ when $A$ is equal to

\begin{aligned} \frac{4}{3 - 4 σ} & for 1 / 2 < σ \leq 5 / 8; \\ \frac{10}{5 - 6 σ} & for 5 / 8 < σ \leq 35 / 54; \\ \frac{19}{6 - 6 σ} & for 35 / 54 < σ \leq 41 / 60; \\ \frac{2112}{859 - 948 σ} & for 41 / 60 < σ \leq 3 / 4; \\ \frac{12408}{4537 - 4890 σ} & for 3 / 4 \leq σ \leq 5 / 6; \\ \frac{4324}{1031 - 1044 σ} & for 5 / 6 \leq σ \leq 7 / 8; \\ \frac{98}{31 - 32 σ} & for 7 / 8 \leq σ \leq 0.91591 \dots; \\ \frac{24 σ - 9}{(4 σ - 1) (1 - σ)} & for 0.91591 \dots \leq σ < 1. \end{aligned}

Additionally, for $σ = 2 / 3$ one can take $A = 9.6187 \dots$ , for $σ = 7 / 10$ one can take $A = 11$ , and for $σ = 5 / 7$ one can take $A = 12$ .

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Corollary 11.34Zero density estimates for

σ

close to

3 / 4

[ 110 , Theorem 11.5 ] One has $A (σ) \leq \frac{3}{7 σ - 4}$ for $3 / 4 \leq σ \leq 10 / 13$ , and $A (σ) \leq \frac{9}{8 σ - 2}$ for $10 / 13 \leq σ \leq 1$ .

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Corollary 9.13Mixed moments and exponent pairs

If $(k, ℓ)$ is an exponent pair with $k > 0$ , then

M (1 / 2, 6, \geq h) \leq 1

and

M (1 / 2, \frac{2 (1 + 2 k + 2 ℓ)}{k}, \leq h) \leq \frac{k + ℓ}{k}

where

h := \frac{ℓ}{2 + 4 l - 2 k} .

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Lemma 11.261980 Ivic zero density bound

[ 109 ] , [ 110 , Theorem 11.2 ] We have

A (σ) \leq \frac{4}{2 σ + 1}

for $17 / 18 \leq σ \leq 1$ , and

A (σ) \leq \frac{24}{30 σ - 11}

for $155 / 174 \leq σ \leq 17 / 18$ .

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Corollary 11.281971 Montgomery zero density bound

[ 170 ] , [ 110 , Theorem 11.3 ] For any $9 / 10 \leq σ \leq 1$ and $1 / 2 \leq α \leq 1$ one has

A (σ) (1 - σ) \leq \frac{7}{6} μ (5 σ - 4) .

In particular, for $152 / 155 \leq σ \leq 1$ , one has

A (σ) \leq min (35 / 36, 1600 (1 - σ)^{1 / 2}) .

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Theorem 11.17Jutila zero density theorem

[ 128 ] The zero density hypothesis is true for $σ \geq 11 / 14$ .

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Theorem 7.16Jutila large values theorem

For any integer $k \geq 1$ , one has

LV (σ, τ) \leq max (2 - 2 σ, τ + (4 - 2 / k) - (6 - 2 / k) σ, τ + (6 - 8 σ) k) .

Thus for instance with $k = 2$ we have

LV (σ, τ) \leq max (2 - 2 σ, τ + 3 - 5 σ, τ + 12 - 16 σ)

and with $k = 3$ we have

LV (σ, τ) \leq max (2 - 2 σ, τ + \frac{10 - 16 σ}{3}, τ + 18 - 24 σ) .

In particular, the Montgomery conjecture holds for

τ \leq min ((4 - 2 / k) σ - (2 - 2 / k), (8 k - 2) σ - 6 k + 2) .

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Proposition 11.37

[ 130 , Theorems 6, 7 ] One has $A (σ) \leq \frac{3}{2 σ}$ for $σ \geq 23 / 29$ , and

A (σ) \leq max (\frac{36}{138 σ - 89}, \frac{114 σ - 79}{(1 - σ) (138 σ - 89)})

for $127 / 168 \leq σ \leq 107 / 138$ .

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Lemma 10.32Kerr large values theorem

[ 130 , Theorem 2 ] Let $3 / 4 < σ \leq 1$ , $0 \leq τ \leq 3 / 2$ , and $0 \leq ρ \leq LV (σ, τ), 1$ be fixed. Then for any fixed $α \geq 0$ , one has

$ρ \leq max (2 - 2 σ + α, 2 τ + 4 - 8 σ - α, τ / 3 + 16 / 3 - 20 σ / 3 + α / 3, 2 τ / 3 + 9 - 12 σ) .$
[ 130 , Theorem 3 ] Under the same hypotheses as (i), we have for any fixed integer $k \geq 2$ obeying $- α < 4 k σ - (τ + 3 k - 1)$ and $- α < 1 + \frac{1}{k - 1} - τ$ that

$ρ \leq max (2 - 2 σ + α, τ / 3 + (3 k + 4) / 3 - (4 k + 4) σ / 3 + α / 3) .$
[ 130 , Theorem 4 ] If $25 / 32 < σ \leq 1$ , $τ \geq 0$ , and $0 \leq ρ \leq LV (σ, τ), 1, 4 - 2 τ$ are fixed, then for any fixed $α$ with

$26 - 32 σ - τ < - α < 16 σ - 11 - τ$

one has

$ρ \leq max (2 - 2 σ + α, 2 τ + 4 - 8 σ - α, - τ + 8 - 8 σ + 2 α, 10 - 12 σ + 2 α / 3) .$
[ 130 , Theorem 5 ] If $1 / 2 < σ \leq 1$ , $0 \leq τ \leq 3 / 2$ , and $0 \leq ρ \leq LV (σ, τ), 1$ are fixed, then for any fixed $α$ with

$- α < - τ + 8 σ - 5$

one has

$ρ \leq max (2 - 2 σ + α, 4 τ / 3 + 23 / 3 - 12 σ - 2 α / 3, 2 τ / 3 + 14 / 3 - 20 / 3) .$

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Lemma 3.1

L^{2}

integral estimate

Let $ξ_{1}, \dots, ξ_{R}$ be real numbers that are $1 / N$ -separated. Then for any interval $I$ of length $T$ , and any sequence $a_{1}, \dots, a_{R}$ of complex numbers one has

\int_{I} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} d t = (T + O (N)) \sum_{r = 1}^{R} | a_{r} |^{2} .

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Theorem 7.9

L^{2}

mean value theorem

For any fixed $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ one has

LV (σ, τ) \leq max (2 - 2 σ, 1 + τ - 2 σ) .

In particular, the Montgomery conjecture 2 holds for $τ \leq 1$ .

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Definition 7.1Large value pattern

A large value pattern is a tuple $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ , where $N > 1$ and $T, V > 0$ are real numbers, $a_{n}$ is a $1$ -bounded sequence on $[N, 2 N]$ , $J$ is an interval of length $T$ , and $W$ is a $1$ -separated subset of $J$ such that

| \sum_{n \in [N, 2 N]} a_{n} n^{- i t} | \geq V

for all $t \in W$ .

A Zeta large value pattern is a large value pattern in which $J = [T, 2 T]$ and $a_{n} = 1_{I} (n)$ for some interval $I \subset [N, 2 N]$ .

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Corollary 8.8

Assuming the Lindelof hypothesis, one has ${LV}_{ζ} (σ, τ) = - \infty$ whenever $σ > 1 / 2$ and $τ \geq 1$ .

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Theorem 11.12

The Lindelof hypothesis implies the density hypothesis, and also that $A (σ) \leq 0$ for $3 / 4 < σ \leq 1$ .

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Theorem 5.12Exponent pairs on the line of symmetry

$(k, k + 1 / 2)$ is an exponent pair for

$k = 9 / 56$ [ 97 , Theorem 1 ] ;
$k = 89 / 560$ [ 227 , Theorem 6 ] ;
$k = 17 / 108$ [ 95 , p. 467 ] ;
$k = 89 / 570$ [ 91 , p. 40 ] ;
$k = 32 / 205$ [ 94 , Theorem 1 ] ;
$k = 13 / 84$ [ 17 , p. 206 ] .

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Lemma 7.3Asymptotic form of large value exponent

Let $1 / 2 \leq σ \leq 1$ , $τ \geq 0$ , and $ρ \geq 0$ be fixed. Then the following are equivalent:

$LV (σ, τ) \leq ρ$ .
For every (fixed) $ε > 0$ there exists $C, δ > 0$ such that if $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is a large value pattern with $N \geq C$ and $N^{τ - δ} \leq T \leq N^{τ + δ}$ , $N^{σ - δ} \leq V \leq N^{σ + δ}$ , then one has

$| W | \leq C N^{ρ + ε} .$

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Lemma 7.4Basic properties

(Monotonicity in $σ$ ) For any $τ \geq 0$ , $σ \mapsto LV (σ, τ)$ is upper semicontinuous and monotone non-increasing.
(Huxley subdivision) For any $1 / 2 \leq σ \leq 1$ and $τ^{'} \geq τ$ one has

$LV (σ, τ) \leq LV (σ, τ^{'}) \leq LV (σ, τ) + τ^{'} - τ .$

In particular, $τ \mapsto LV (σ, τ)$ is Lipschitz continuous.
( $τ = 0$ endpoint) One has $LV (σ, 0) = 0$ for all $1 / 2 \leq σ \leq 1$ , and hence by (ii) $0 \leq LV (σ, τ) \leq τ$ for all $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ .

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Definition 7.2Large value exponent

Let $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ be fixed. We define $LV (σ, τ)$ to be the least fixed quantity for which the following claim is true: whenever $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is a large value pattern with $N > 1$ unbounded, $T = N^{τ + o (1)}$ , and $V = N^{σ + o (1)}$ , then

| W | ≪ N^{LV (σ, τ) + o (1)} .

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Definition 10.6Large value energy region

The large value energy region $E \subset R^{5}$ is defined to be the set of all fixed tuples $(σ, τ, ρ, ρ^{*}, s)$ with $1 / 2 \leq σ \leq 1$ , $τ, ρ, ρ^{'} \geq 0$ , such that there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N > 1$ unbounded, $V = N^{σ + o (1)}$ , $T = N^{τ + o (1)}$ , $V = N^{σ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ and $S (N, W) = N^{s + o (1)}$ .

We define the zeta large value energy region $E_{ζ} \subset R^{5}$ similarly, but where now $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is required to be a zeta large value pattern.

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Lemma 7.5Lower bound

For any $1 / 2 < σ \leq 1$ and $τ \geq 0$ , one has $LV (σ, τ) \geq min (2 - 2 σ, τ)$ , while for $σ = 1 / 2$ one has $LV (σ, τ) = τ$ .

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Lemma 10.10Non-asymptotic form of large value energy region

Let $1 / 2 \leq σ \leq 1$ , $τ \geq 0$ , $ρ, ρ^{*} \geq 0$ , and $s \in R$ be fixed. Then the following are equivalent:

$(σ, τ, ρ, ρ^{*}, s) \in E$ .
For every $ε > 0$ and $C > 0$ , there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N \geq C$ , $N^{τ - δ} \leq T \leq N^{τ + δ}$ , $N^{σ - δ} \leq V \leq N^{σ + δ}$ , $N^{ρ - ε} \leq | W | \leq N^{ρ + ε}$ , $N^{ρ^{*} - ε} \leq E_{1} (W) \leq N^{ρ^{*} + ε}$ , and $N^{s - ε} \leq S (N, W) \leq N^{s + ε}$ .

Similarly with $E$ replaced by $E_{ζ}$ , and with $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ required to be a zeta large value pattern.

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Lemma 10.11Basic properties

(Monotonicity in $σ$ ) If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then $(σ^{'}, τ^{'}, ρ, ρ^{*}, s) \in E$ for all $1 / 2 \leq σ^{'} \leq σ$ and $τ^{'} \geq τ$ .
(Subdivision) If $(σ, τ, ρ, ρ^{*}, s) \in E$ and $0 \leq τ^{'} \leq τ$ , then amongst all tuples $(σ, τ^{'}, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ with $ρ^{'} \leq ρ$ , $(ρ^{'})^{*} \leq ρ^{*}$ , and $s^{'} \leq s$ , there exists a tuple with

$ρ \leq ρ^{'} + τ - τ^{'};$

there exists a tuple with

$ρ^{*} \leq ρ^{'} + 3 min (ρ - ρ^{'}, τ - τ^{'});$

and there exists a tuple with

$s \leq s^{'} + 2 min (ρ - ρ^{'}, τ - τ^{'}) .$

(But it may not be the same tuple that satisfies all three properties.)
(Trivial bounds) If $(σ, τ, ρ, ρ^{*}, s) \in E$ , one has

$2 ρ, 4 ρ - τ \leq ρ^{*} \leq 3 ρ .$

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Lemma 10.13Monotonicity criterion

Let $E_{1}$ be the intersection of sets $E_{i}$ , each of the form

\begin{array}{r} {(σ, τ, ρ, ρ^{*}, s) \in R^{5} : ρ \leq f_{1} (ρ^{*}, s), ρ^{*} \leq f_{2} (ρ, s), s \leq f_{3} (ρ, ρ^{*})} \end{array}

for some monotonically increasing functions $f_{1}, f_{2}, f_{3}$ (that possibly also depend on $σ$ and $τ$ ).

Suppose amongst all tuples $(σ, τ, ρ^{'}, (ρ^{*})^{'}, s^{'}) \in E_{1}$ with $ρ^{'} \leq ρ$ , $(ρ^{*})^{'} \leq ρ^{*}$ and $s^{'} \leq s$ , there exists a tuple with $ρ^{'} = ρ$ , a tuple with $(ρ^{*})^{'} = ρ^{*}$ and a tuple with $s^{'} = s$ (not necessarily the same tuple each time). Then, $(σ, τ, ρ, ρ^{*}, s) \in E_{1}$ .

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Lemma 8.9Value at

σ = 1 / 2

One has ${LV}_{ζ} (1 / 2, τ) = τ$ for all $τ \geq 1$ .

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Corollary 9.6Fourth moment bound

One has ${LV}_{ζ} (σ, τ) \leq τ - 4 (σ - 1 / 2)$ for all $1 / 2 \leq σ \leq 1$ and $τ \geq 2$ .

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Lemma 8.2Asymptotic form of large value exponent at zeta

Let $1 / 2 \leq σ \leq 1$ , $τ \geq 0$ , and $ρ \geq 0$ be fixed. Then the following are equivalent:

${LV}_{ζ} (σ, τ) \leq ρ$ .
For every $ε > 0$ there exists $C, δ > 0$ such that if $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is a zeta large value pattern with $N > C$ , $N^{τ - δ} \leq T \leq N^{τ + δ}$ , and $N^{σ - δ} \leq V \leq N^{σ + δ}$ , then one has

$| W | \leq C N^{ρ + ε} .$

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Lemma 8.3Basic properties

(Monotonicity in $σ$ ) For any $τ \geq 0$ , $σ \mapsto {LV}_{ζ} (σ, τ)$ is upper semicontinuous and monotone non-increasing.
(Trivial bound) For any $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ , we have ${LV}_{ζ} (σ, τ) \leq τ$ .
(Domination by large values) We have ${LV}_{ζ} (σ, τ) \leq LV (σ, τ)$ for all $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ .
(Reflection) For $1 / 2 \leq σ \leq 1$ and $τ > 1$ , one has

$sup_{σ \leq σ^{'} \leq 1} {LV}_{ζ} (\frac{1}{2} + \frac{1}{τ - 1} (σ^{'} - \frac{1}{2}), \frac{τ}{τ - 1}) + \frac{1}{τ - 1} (σ^{'} - σ) = \frac{1}{τ - 1} sup_{σ \leq σ^{'} \leq 1} ({LV}_{ζ} (σ^{'}, τ) + σ^{'} - σ) .$

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Definition 8.1Large value zeta exponent

Let $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ be fixed. We define ${LV}_{ζ} (σ, τ) \in [- \infty, + \infty)$ to be the least (fixed) exponent for which the following claim is true: if $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is a zeta large value pattern with $N$ is unbounded, $T = N^{τ + o (1)}$ , and $V = N^{σ + o (1)}$ , then $| W | ≪ N^{ρ + o (1)}$ .

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Corollary 8.7

If $(k, ℓ)$ is an exponent pair, then ${LV}_{ζ} (σ, τ) = - \infty$ whenever $1 / 2 \leq σ \leq 1$ , $τ \geq 0$ are fixed quantities with $σ > k τ + ℓ - k$ .

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Lemma 8.4Characterization of negative infinite value

Let $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ be fixed. Then the following are equivalent:

${LV}_{ζ} (σ, τ) = - \infty$ .
${LV}_{ζ} (σ, τ) < 0$ .
There exists a fixed $ε > 0$ such that if $N$ is unbounded and $I$ is a subinterval of $[N, 2 N]$ , then one has

$\sum_{n \in I} n^{- i t} ≪ N^{σ - ε + o (1)}$

whenever $| t | = N^{τ + o (1)}$ .

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Corollary 8.6

If $τ > 0$ and $1 / 2 \leq σ_{0} \leq 1$ are fixed, then ${LV}_{ζ} (σ, τ) = - \infty$ whenever $σ > σ_{0} + τ μ (σ_{0})$ .

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Lemma 8.10Value at

τ < 1

If $0 \leq τ < 1$ , then ${LV}_{ζ} (σ, τ)$ is equal to $- \infty$ for $σ > 1 - τ$ and equal to $τ$ for $σ \leq 1 - τ$ .

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Definition 10.9

For any fixed $1 / 2 \leq σ \leq 1, τ \geq 0$ , we define

{LV}^{*} (σ, τ) := sup {ρ^{*} : (σ, τ, ρ, ρ^{*}, s) \in E}

and

{LV}_{ζ}^{*} (σ, τ) := sup {ρ^{*} : (σ, τ, ρ, ρ^{*}, s) \in E_{ζ}} .

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Theorem 9.10Moment bounds for

σ = 1 / 2

[ 218 , Theorems 2.1, 2.2 ] We have

M (1 / 2, A) \leq {\begin{cases} (16 A + 35) / 114, & \frac{866}{65} \leq A < 14, \\ (176677 A + 358428) / 1246476, & 14 \leq A < \frac{122304}{7955} = 15.37 \dots, \\ (779 A + 1398) / 5422, & \frac{122304}{7955} \leq A < \frac{910020}{58699} = 15.50 \dots, \\ 3 (1661 A + 2856) / 34532, & \frac{910020}{58699} \leq A < \frac{9604}{593} = 16.19 \dots, \\ (405277 A + 677194) / 2800950, & \frac{9604}{593} \leq A < \frac{629068}{35731} = 17.60 \dots, \\ (40726597 A + 64268678) / 280113282, & \frac{629068}{35731} \leq A < \frac{13789}{709} = 19.44 \dots, \\ 3 (46 A + 49) / 926, & \frac{13789}{709} \leq A < \frac{204580}{10333} = 19.79 \dots, \\ (3475 A + 3236) / 23168, & \frac{204580}{10333} \leq A < \frac{4252}{195} = 21.80 \dots, \\ 7 (39945 A + 33704) / 1857036, & \frac{4252}{195} \leq A < \frac{812348}{30267} = 26.83 \dots, \\ (37 A + 24) / 244, & \frac{812348}{30267} \leq A < \frac{440}{13} = 33.84 \dots, \\ (31 A - 24) / 196, & \frac{440}{13} \leq A < \frac{203087}{4742} = 42.82 \dots, \\ 7 (31519 A - 33704) / 1385180, & \frac{203087}{4742} \leq A < \frac{3516129}{65729} = 53.49 \dots, \\ 1 + 13 (A - 6) / 84, & \frac{3516129}{65729} \leq A . \end{cases}

and also

M (1 / 2, 12 + δ) \leq 2 + \frac{δ}{8} + \frac{3 \sqrt{510}}{7568} δ^{3 / 2}, 0 < δ \leq \frac{86}{65} .

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Lemma 9.5

If $1 / 2 \leq σ_{0} \leq 1$ and $A \geq 1$ , then

M (σ_{0}, A) = sup_{τ \geq 2; σ \geq 1 / 2} (A (σ - σ_{0}) + {LV}_{ζ} (σ, τ)) / τ .

In particular, one has

{LV}_{ζ} (σ, τ) \leq τ M (σ_{0}, A) - A (σ - σ_{0})

whenever $σ \geq 1 / 2$ and $τ \geq 2$ .

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Lemma 9.12Mixed moments and large values of zeta

If $1 / 2 \leq σ_{0} \leq 1$ , $A \geq 1$ , and $h \geq 0$ are fixed, then

M (σ_{0}, A, \geq h) \leq sup_{τ \geq 2; σ \geq 1 / 2, h τ} (A (σ - σ_{0}) + {LV}_{ζ} (σ, τ)) / τ .

and

M (σ_{0}, A, \leq h) \leq sup_{τ \geq 2; σ \leq 1 / 2, h τ} (A (σ - σ_{0}) + {LV}_{ζ} (σ, τ)) / τ .

That is to say, any bound of the form

{LV}_{ζ} (σ, τ) \leq M τ - A (σ - σ_{0})

whenever $τ \geq 2$ and $σ \geq 1 / 2, h τ$ , gives rise to a bound

M (σ_{0}, A, \geq h) \leq M .

Similarly for $M (σ_{0}, A, \geq h)$ , in which we replace the condition $σ \geq h τ$ by $σ \leq h τ$ .

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Lemma 9.4

One has $M (1 / 2, A) = 1$ for all $0 \leq A \leq 4$ .

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Lemma 14.9

Let $k \geq 2$ be an integer. If $M (σ, k) = 1$ then $α_{k} \leq σ$ .

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Definition 9.11Mixed moments

For fixed $1 / 2 \leq σ \leq 1$ , $A \geq 0$ , and $h \geq 0$ , let $M (σ, A, \geq h)$ be the least (fixed) exponent for which the bound

\int_{0 \leq t \leq T : | ζ (σ + i t) | \geq T^{h}} | ζ (σ + i t) |^{A} d t ≪ T^{M (σ, A, h) + o (1)}

holds for unbounded $T$ . Similarly, let $M (σ, A, \leq h)$ be the least exponent for which

\int_{0 \leq t \leq T : | ζ (σ + i t) | < T^{h}} | ζ (σ + i t) |^{A} d t ≪ T^{M (σ, A, h) + o (1)}

holds for unbounded $T$ .

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Corollary 9.3Relationship with Lindelöf hypothesis

If the Lindelöf hypothesis holds, then $M (σ, A) = 1 + max (1 / 2 - σ, 0) A$ for all $σ \in R$ and $A \geq 0$ . Conversely, if $M (1 / 2, A) = 1$ for arbitrarily large $A \geq 0$ , then the Lindelöf hypothesis is true.

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Theorem 7.10Montgomery large values theorem

If $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ is such that

sup_{1 \leq τ^{'} \leq τ} β (1 / τ^{'}) τ^{'} < 2 σ - 1

(this condition is vacuous for $τ < 1$ ) then the Montgomery conjecture 2 holds for this choice of parameters.

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Lemma 7.7Subdivision and the Montgomery conjecture

If $σ$ is fixed, and the Montgomery conjecture holds for all fixed $τ < τ_{0}$ , then

LV (σ, τ) \leq max (2 - 2 σ, τ - τ_{0} + 2 - 2 σ)

for all fixed $τ \geq 0$ .

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Theorem 11.11

The Montgomery conjecture implies the density hypothesis.

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Proposition 6.11Conjectured value of

μ

We have the lower bound

μ (σ) \geq max (0, \frac{1}{2} - σ)

for all $σ \in R$ , and equality holds everywhere in 1 if and only if the Lindelöf hypothesis holds.

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Theorem 6.12Historical bounds

The upper bounds on $μ (σ)$ given by Table 6.1 are known.

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Corollary 8.13Converting a bound on

μ

to a large values theorem

If $1 / 2 \leq σ \leq 1$ , $σ^{'} \leq 1$ , and $τ \geq 0$ are fixed, then

LV (σ, τ) \leq max (2 - 2 σ, 2 - 2 σ + τ - \frac{2 σ - 1 - σ^{'}}{μ (σ^{'})}) .

In particular, the Montgomery conjecture holds for $τ \leq \frac{2 σ - 1 - σ^{'}}{μ (σ^{'})}$ .

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Theorem 6.13

[ 218 , Theorems 2.4-2.6 ] We have

μ (σ) \leq {\begin{cases} (31 - 36 σ) / 84, & \frac{1}{2} \leq σ < \frac{88225}{153852} = 0.5734 \dots, \\ (220633 - 251324 σ) / 620612, & \frac{88225}{153852} \leq σ < \frac{521}{796} = 0.6545 \dots, \\ (1333 - 1508 σ) / 3825, & \frac{521}{796} \leq σ < \frac{53141}{76066} = 0.6986 \dots, \\ (405 - 454 σ) / 1202, & \frac{53141}{76066} \leq σ < \frac{3620}{5119} = 0.7071 \dots, \\ (49318855 - 52938216 σ) / 170145110, & \frac{3620}{5119} \leq σ < \frac{52209}{69128} = 0.7552 \dots, \\ (471957 - 502648 σ) / 1682490, & \frac{52209}{69128} \leq σ < \frac{1389}{1736} = 0.8001 \dots, \\ (2841 - 3016 σ) / 10316, & \frac{1389}{1736} \leq σ < \frac{134765}{163248} = 0.8255 \dots, \\ (859 - 908 σ) / 3214, & \frac{134765}{163248} \leq σ < \frac{18193}{21906} = 0.8305 \dots, \\ 5 (8707 - 9067 σ) / 180277, & \frac{18193}{21906} \leq σ < \frac{249}{280} = 0.8892 \dots, \\ (29 - 30 σ) / 130, & \frac{249}{280} \leq σ \leq \frac{9}{10} . \end{cases}

Furthermore, for $1 / 2 \leq σ \leq 1$ , we have

μ (σ) \leq \frac{2}{13} \sqrt{10} (1 - σ)^{3 / 2} = 0.4865 \dots (1 - σ)^{3 / 2},

and

μ (σ) \leq \frac{2}{3^{3 / 2}} (1 - σ)^{3 / 2} + \frac{103}{300} (1 - σ)^{2}, \frac{117955}{118272} \leq σ \leq 1.

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Lemma 18.5Relation to growth exponents of zeta

Suppose $f (t) > 0$ and $0 < g (t) \leq 1$ are real-valued functions for $t \geq 0$ , with $f (t)$ non-decreasing and tending to infinity with $t$ , and $g (t)$ non-increasing. Suppose further that $f (t) / g (t) = o (\exp (f (t)))$ . If

ζ (σ + i t) ≪ \exp (f (t)) (1 - g (t) \leq σ \leq 2, t \geq 0)

then $ζ (σ + i t) \neq 0$ for

σ \geq 1 - A \frac{g (2 t + 1)}{f (2 t + 1)}

where $A > 0$ is an absolute constant.

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Theorem 5.22New exponent pairs

The following are exponent pairs:

(\frac{89}{1282}, \frac{997}{1282}), (\frac{652397}{9713986}, \frac{7599781}{9713986}), (\frac{10769}{351096}, \frac{609317}{702192}), (\frac{89}{3478}, \frac{15327}{17390}) .

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Definition 4.1Phase function

A phase function is a (variable) smooth function $F : [1, 2] \to R$ . A phase function $F$ will be called a model phase function if there exists a fixed exponent $σ > 0$ with the property that

F^{(p + 1)} (u) - \frac{d^{p}}{d u^{p}} u^{- σ} = o (1)

for all (variable) $u \in [1, 2]$ and all fixed $p \geq 0$ , where $F^{(p + 1)}$ denotes the $(p + 1)^{st}$ derivative of $F$ .

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Lemma 14.10Piltz bound

For $k \geq 2$ , one has

α_{k} \leq 1 - \frac{1}{k} .

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Theorem 11.35Pintz zero density theorem

[ 185 , Theorem 1 ] If $k \geq 4$ , $ℓ \geq 3$ are integers and $σ = 1 - η$ is such that

\frac{1}{k (k + 1)} \leq η < \frac{1}{k (k - 1)}

and

\frac{1}{2 ℓ (ℓ + 1)} \leq η < \frac{1}{2 ℓ (ℓ - 1)}

then

A (σ) \leq max (\frac{3}{ℓ (1 - 2 (ℓ - 1) η)}, \frac{4}{k (1 - (k - 1) η)}) .

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Definition 13.1Prime number theorem in short interval exponents

We let $θ_{PNT}$ denote the least exponent with the following property: if $ε > 0$ is fixed, and $x$ is unbounded, then

$\sum_{x \leq n < x + y} Λ (n) = y + o (y)$

whenever $x^{θ_{PNT} + ε} \leq y \leq x^{1 - ε}$ .
We let $θ_{PNT - AA}$ denote the least exponent with the following property: if $ε > 0$ is fixed, and $X$ is unbounded, then we have

$\int_{X}^{2 X} | \sum_{x \leq n < x + y} Λ (n) - y | d x = o (X y)$

whenever $X^{θ_{PNT - AA} + ε} \leq y \leq X^{1 - ε}$ .
We let $θ_{gap}$ denote the least exponent such that, if $p_{n}$ denotes the $n^{th}$ prime, that

$p_{n + 1} - p_{n} ≪ n^{θ_{gap} + o (1)} = p_{n}^{θ_{gap} + o (1)}$

as $n \to \infty$ .
We let $θ_{gap, 2}$ denote the least exponent such that

$\sum_{p_{n} \leq x} (p_{n + 1} - p_{n})^{2} ≪ x^{θ_{gap, 2} + o (1)}$

as $x \to \infty$ .
We let $θ_{gap - AA}$ denote the least exponent such that for every $ε > 0$ , the intervals $[n, n^{θ_{gap - AA} + ε}]$ contain a prime for a density $1$ set of natural numbers $n$ .

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Lemma 13.2Trivial bounds

We have

0 \leq θ_{gap - AA} \leq θ_{PNT - AA}, θ_{gap} \leq θ_{PNT} \leq 1

and $1 \leq θ_{gap, 2} \leq 1 + θ_{gap}$ .

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Lemma 10.12Raising to a power

If $(σ, τ, ρ, ρ^{*}, s) \in E$ , and $k \geq 1$ , then amongst all tuples $(σ, τ / k, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ with $ρ^{'} \leq ρ / k$ , $(ρ^{'})^{*} \leq ρ^{*} / k$ , and $s^{'} \leq s / k$ , there exists a tuple with $ρ^{'} = ρ / k$ , there exists a tuple with $(ρ^{'})^{*} = ρ^{*} / k$ , and there exists a tuple with $s^{'} = s / k$ . (These may be three different tuples.)

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Lemma 7.8Raising to a power

For any $1 / 2 \leq σ \leq 1$ , $τ \geq 0$ , and natural number $k$ , one has

LV (σ, k τ) \leq k LV (σ, τ) .

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Proposition 13.4Zero density theorems and prime gaps

Let

‖ A ‖_{\infty} := sup_{1 / 2 \leq σ \leq 1} A (σ) .

Then

θ_{PNT} \leq 1 - \frac{1}{‖ A ‖_{\infty}}

and

θ_{PNT - AA} \leq 1 - \frac{2}{‖ A ‖_{\infty}} .

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Lemma 15.2

One has $θ_{Pythag} \leq 1 / 4$ .

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Lemma 15.4

Assuming RH, one has $θ_{Pythag} \leq 71 / 316$ .

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Definition 15.1Pythagorean triple exponent

Let $θ_{Pythag}$ be the least exponent for which one has

P (N) = c N^{1 / 2} - c^{'} N^{1 / 3} + N^{θ_{Pythag} + o (1)}

for unbounded $N$ and some fixed $c, c^{'}$ , where $P (N)$ is the number of primitive Pythagorean triples of area no greater than $N$ .

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Theorem 10.15Reflection principle

[ 110 , § 11.5 ] If $(σ, τ, ρ, ρ^{*}, s) \in E$ with $σ \geq 3 / 4$ and $τ > 1$ , then for any integer $k \geq 1$ , either $ρ \leq 2 - 2 σ$ , or there exists $0 < α \leq k (τ - 1)$ and $(σ, τ / α, ρ / α, ρ^{*} / α, s^{'} / α) \in E$ such that

ρ \leq min (2 - 2 σ, k (3 - 4 σ) / 2 + s^{'} - 1) .

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Theorem 4.19Sargos 2003 bound

For any $α > 0$ one has

β (α) \leq max (α + \frac{1 - 8 α}{204}, - \frac{95 (1 - 8 α)}{204})

and

β (α) \leq max (α + \frac{7 (1 - 9 α)}{2640}, - \frac{1001 (1 - 9 α)}{2640}) .

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Theorem 4.151995 Sargos bound

[ 202 , Th é or è me 2.4, Lemme 2.6 ] For any $0 \leq α \leq 1$ , one has

β (α) \leq max (α + \frac{3 (1 - 4 α)}{40}, \frac{7}{8} α, \frac{1}{3} α - \frac{1 - 4 α}{6}, 0)

and

β (α) \leq max (α + \frac{1 - 4 α}{14}, \frac{5}{6} α, \frac{1}{3} α - \frac{1 - 4 α}{6}, 0) .

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Theorem 5.18Sargos

C

-process

[ 203 , Theorem 5 ] If $(k, ℓ)$ is an exponent pair, then so is

(\frac{k}{12 (1 + 4 k)}, \frac{11 (1 + 4 k) + ℓ}{12 (1 + 4 k)}) .

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Theorem 5.19Sargos

D

-process

[ 202 , Theorem 7.1 ] If $(k, ℓ)$ is an exponent pair, then one has

β (α) \leq max (k_{1} + α (ℓ_{1} - k_{1}), \frac{1}{12} + \frac{2}{3} α)

for all $0 \leq α \leq 1$ , where $(k_{1}, ℓ_{1}) = D (k, ℓ)$ is the pair

D (k, ℓ) := (\frac{5 k + ℓ + 2}{8 (5 k + 3 ℓ + 2)}, \frac{29 k + 21 ℓ + 10}{8 (5 k + 3 ℓ + 2)}) .

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Theorem 11.13Ingham’s first bound

[ 103 ] (See also [ 217 ] ) For any $1 / 2 < σ < 1$ , we have

A (σ) \leq 2 + 4 μ (1 / 2) .

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Theorem 11.14Ingham’s second bound

[ 104 ] For any $1 / 2 < σ < 1$ , one has $A (σ) \leq \frac{3}{2 - σ}$ .

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Theorem 19.3Brun-Titchmarsh theorem under GRH (1929) [ 213 ]

Under the Generalized Riemann Hypothesis (GRH), if $q < x$ , then

π (x; q, a) = \frac{Li (x)}{φ (q)} + O (x^{1 / 2} \log x) .

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Lemma 10.7Trivial containment

We have $E_{ζ} \subset E$ .

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Proposition 13.11Trivial bounds on large gaps

One has $θ_{gap, >} \leq θ_{gap, \geq}$ . If $θ_{gap} < 1 / 2$ , then $θ_{gap, >} = - \infty$ . In general, we have

max (1 / 2, θ_{gap}) \leq max (1 / 2, θ_{gap, >})

and $θ_{gap, >} \leq 1$ . Also $θ_{gap, >} \leq θ_{gap, 2} - 1 / 2$ .

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Theorem 5.20

[ 218 , Lemma 1.1 ] The following are exponent pairs:

\begin{aligned} (k_{1}, ℓ_{1}) & := (\frac{4742}{38463}, \frac{35731}{51284}) \\ (k_{2}, ℓ_{2}) & := (\frac{18}{199}, \frac{593}{796}) \\ (k_{3}, ℓ_{3}) & := (\frac{2779}{38033}, \frac{58699}{76066}) \\ (k_{4}, ℓ_{4}) & := (\frac{715}{10238}, \frac{7955}{10238}) . \end{aligned}

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Proposition 5.8Van der Corput

A

-process

If $(k, ℓ)$ is an exponent pair, then so is

A (k, ℓ) := (\frac{k}{2 k + 2}, \frac{ℓ}{2 k + 2} + \frac{1}{2}) .

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Proposition 5.9Van der Corput

B

-process

If $(k, ℓ)$ is an exponent pair, then so is

B (k, ℓ) := (ℓ - \frac{1}{2}, k + \frac{1}{2}) .

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Proposition 5.10Classical van der Corput exponent pairs

For any natural number $k \geq 2$ ,

A^{k - 2} B (0, 1) = (\frac{1}{2^{k} - 2}, 1 - \frac{k - 1}{2^{k} - 2})

is an exponent pair. In particular,

(\frac{1}{2}, \frac{1}{2}), (\frac{1}{6}, \frac{2}{3}), (\frac{1}{14}, \frac{11}{14})

are exponent pairs.

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Corollary 4.8Optimizing the van der Corput inequality

For any $α > 0$ one has

β (α) \leq inf_{k \in N : k \geq 2} α + \frac{1 - k α}{2^{k} - 2} .

Thus for instance

β (α) \leq min (\frac{1}{2}, \frac{1 + 3 α}{6}, \frac{10 α + 1}{14}) .

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Lemma 4.6Van der Corput

A

process for

β

If $0 \leq α \leq 2 / 3$ and $h \geq 0$ then

2 β (α) \leq max (2 α - h, 2 h, α - h + sup_{2 α - 1 \leq h^{'} \leq h} ((h^{'} + 1 - α) β (\frac{α}{h^{'} + 1 - α}) + h^{'})) .

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Lemma 14.11Voronoi, Landau bound

For $k \geq 2$ , one has

α_{k} \leq 1 - \frac{2}{k + 1} .

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Theorem 19.4Walfisz (1936) [ 224 ]

Fix $B \geq 0$ and suppose $q \leq (\log x)^{B}$ . Then there exists $A = A (B) > 0$ such that

π (x; q, a) = \frac{Li (x)}{φ (q)} + O (x \exp (- A \sqrt{\log x})) .

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Lemma 10.5

If $W \subset [- T, T]$ is $1$ -separated and $1 \leq N ≪ T^{O (1)}$ , then one has

S_{4} (N, W) ≪ T^{o (1)} u^{2} S (N, U) + T^{- 100}

for some $1 \leq u ≪ | W |$ and $1$ -separated subset $U$ of $[- 2 T, 2 T]$ with

u | U | ≪ | W |^{2}

and

u^{2} | U | ≪ E_{1} (W) .

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Lemma 10.19

If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then there exists $(σ, τ, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ and $0 \leq κ \leq ρ$ such that

κ + ρ^{'} \leq 2 ρ

2 κ + ρ^{'} \leq ρ^{*}

and

ρ^{*} + 2 σ \leq κ + (s + s^{'}) / 2.

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Theorem 12.14

For $165 / 226 \leq σ \leq 42 / 55$ one has

A^{*} (σ) \leq max (\frac{457 - 546 σ}{2 (61 - 58 σ) (1 - σ)}, \frac{5 (18 - 19 σ)}{2 (5 σ + 3) (1 - σ)}) .

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Lemma 11.2Basic properties of

A

$σ \mapsto (1 - σ) A (σ)$ is non-increasing and left-continuous, with $A (1 / 2) = 2$ .
If the Riemann hypothesis holds, then $A (σ) = - \infty$ for all $1 / 2 < σ \leq 1$ .

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Definition 11.1Zero density exponents

For $σ \in R$ and $T > 0$ , let $N (σ, T)$ denote the number of zeroes $ρ$ of the Riemann zeta function with $Re (ρ) \geq σ$ and $| Im (ρ) | \leq T$ .

If $1 / 2 \leq σ < 1$ is fixed, we define the zero density exponent $A (σ) \in [- \infty, \infty)$ to be the infimum of all (fixed) exponents $A$ for which one has

N (σ - δ, T) ≪ T^{A (1 - σ) + o (1)}

whenever $T$ is unbounded and $δ > 0$ is infinitesimal.

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Lemma 11.6Large values from zero density

[ 162 , Theorem 1.2 ] If $τ > 0$ and $1 / 2 \leq σ \leq 1$ are fixed, then

{LV}_{ζ} (σ, τ) / τ \leq max (\frac{1}{2}, sup_{σ \leq σ^{'} \leq 1} A (σ^{'}) (1 - σ^{'}) + \frac{σ^{'} - σ}{2}) .

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Lemma 18.3Basic properties of zero free regions

The following properties hold:

(Symmetry about the real axis) If $ζ (σ + i t) \neq 0$ then $ζ (σ - i t) \neq 0$ .
(Symmetry about the critical line $ℜ s = 1 / 2$ ) For $0 \leq σ \leq 1$ , if $ζ (σ + i t) \neq 0$ then $ζ (1 - σ + i t) \neq 0$ .
(Non vanishing for $ℜ s > 1$ ) If $ℜ s > 1$ then $ζ (s) \neq 0$ .

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Lemma 11.5Zero density from large values

Let $1 / 2 < σ < 1$ . Then

A (σ) (1 - σ) \leq max (sup_{τ \geq 2} {LV}_{ζ} (σ, τ) / τ, \underset{τ \to \infty}{lim sup} LV (σ, τ) / τ) .

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Corollary 11.8

Let $1 / 2 < σ < 1$ and $τ_{0} > 0$ . Then

A (σ) (1 - σ) \leq max (sup_{2 \leq τ < 4 τ_{0} / 3} {LV}_{ζ} (σ, τ) / τ, sup_{2 τ_{0} / 3 \leq τ \leq τ_{0}} LV (σ, τ) / τ) .

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Corollary 11.7

Let $1 / 2 < σ < 1$ and $τ_{0} > 0$ . Then

A (σ) (1 - σ) \leq max (sup_{2 \leq τ < τ_{0}} {LV}_{ζ} (σ, τ) / τ, sup_{τ_{0} \leq τ \leq 2 τ_{0}} LV (σ, τ) / τ)

with the convention that the first supremum is $- \infty$ if it is vacuous (i.e., if $τ_{0} < 2$ ).

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Corollary 11.9

Let $1 / 2 < σ < 1$ and $τ_{0} > 0$ . Suppose that one has the bounds

LV (σ, τ) \leq (3 - 3 σ) \frac{τ}{τ_{0}}

for $2 τ_{0} / 3 \leq τ \leq τ_{0}$ , and

{LV}_{ζ} (σ, τ) \leq (3 - 3 σ) \frac{τ}{τ_{0}}

for $2 \leq τ < 4 τ_{0} / 3$ . Then $A (σ) \leq \frac{3}{τ_{0}}$ .

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Corollary 11.10

Let $1 / 2 < σ < 1$ and $τ_{0} > 0$ . Suppose that one has the bound 7 for $2 \leq τ < 4 τ_{0} / 3$ , and the Montgomery conjecture $LV (σ, τ) \leq 2 - 2 σ$ whenever $0 \leq τ \leq τ_{0} + σ - 1$ . Then $A (σ) \leq \frac{3}{τ_{0}}$ .

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Lemma 18.10Relation to prime number theorem error term

Suppose $ζ (σ + i t) \neq 0$ for $σ \geq 1 - η (t)$ where $η (t)$ is a positive and decreasing function. Then

\sum_{n \leq x} Λ (n) - x ≪ x \exp (- A ω (x)), (x \to \infty)

for an absolute constant $A > 0$ , where

ω (x) := inf_{t \geq 1} (η (t) \log x + \log t) .

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Lemma 11.27Zero density from

μ

bound

[ 170 , Theorem 12.3 ] If $1 / 2 \leq α \leq 1$ and $\frac{α + 1}{2} \leq σ \leq 1$ , then

A (σ) \leq μ (α) \frac{2 (3 σ - 1 - 2 α)}{(2 σ - 1 - α) (σ - α)} .

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Lemma 12.2Basic properties of

A^{*}

We have the trivial bounds

$2 A (σ), 4 A (σ) - \frac{1}{1 - σ} \leq A^{*} (σ) \leq 3 A (σ)$

for any $1 / 2 \leq σ \leq 1$ .
$σ \mapsto (1 - σ) A^{*} (σ)$ is non-increasing, with $A^{*} (1 / 2) = 6$ and $A^{*} (1) = - \infty$ .
If the Riemann hypothesis holds, then $A^{*} (σ) = - \infty$ for all $1 / 2 < σ \leq 1$ .

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Definition 12.1Zero density exponents

For $1 / 2 \leq σ \leq 1$ and $T > 0$ , let $N^{*} (σ, T)$ denote the additive energy $E_{1} (Σ)$ of the imaginary parts of the zeroes $ρ$ of the Riemann zeta function with $Re (ρ) \geq σ$ and $| Im (ρ) | \leq T$ . For fixed $1 / 2 \leq σ \leq 1$ , the zero density exponent $A^{*} (σ) \in [- \infty, \infty)$ is the infimum of all exponents $A^{*}$ for which one has

N^{*} (σ - δ, T) ≪ T^{A^{*} (1 - σ) + o (1)}

for all unbounded $T$ and infinitesimal $δ > 0$ .

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Lemma 12.3Zero density energy from large values energy

Let $1 / 2 < σ < 1$ . Then

A^{*} (σ) (1 - σ) \leq max (sup_{τ \geq 1} {LV}_{ζ}^{*} (σ, τ) / τ, \underset{τ \to \infty}{lim sup} {LV}^{*} (σ, τ) / τ) .

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Corollary 12.4

Let $1 / 2 < σ < 1$ and $τ_{0} > 0$ be fixed. Then

A^{*} (σ) (1 - σ) \leq max (sup_{2 \leq τ < τ_{0}} {LV}_{ζ}^{*} (σ, τ) / τ, sup_{τ_{0} \leq τ \leq 2 τ_{0}} {LV}^{*} (σ, τ) / τ)

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Proposition 12.5Additive energy under the Lindelof hypothesis

Let $1 / 2 \leq σ \leq 1$ be fixed. Then one has

A^{*} (σ) \leq 8 - 4 σ

and $A^{*} (σ) \leq 0$ if $σ > 3 / 4$ .

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Lemma 6.3Convexity

$μ$ is convex.

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Lemma 6.6Convexity bounds

One has $max (0, 1 / 2 - σ) \leq μ (σ) \leq (1 - σ) / 2$ for $0 \leq σ \leq 1$ .

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Lemma 8.11From exponent pairs to zeta large values estimate

[ 110 , Theorem 8.2 ] If $(k, ℓ)$ is an exponent pair with $k > 0$ , then for any $1 / 2 \leq σ \leq 1$ and $τ \geq 0$ one has

{LV}_{ζ} (σ, τ) \leq max (τ - 6 (σ - 1 / 2), \frac{k + ℓ}{k} τ - \frac{2 (1 + 2 k + 2 ℓ)}{k} (σ - 1 / 2)) .

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Lemma 6.4Functional equation

One has $μ (1 - σ) = μ (σ) + σ - 1 / 2$ for all $0 \leq σ \leq 1 / 2$ .

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Definition 6.1Growth rate of zeta

For any fixed $σ \in R$ , let $μ (σ)$ denote the least possible (fixed) exponent for which one has the bound

| ζ (σ + i t) | ≪ | t |^{μ (σ) + o (1)}

for all unbounded $t$ .

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Lemma 6.7Connection with dual exponent pairs

For any $1 / 2 \leq σ \leq 1$ , one has

μ (σ) \leq sup_{0 \leq α \leq 1 / 2} β (α) - α σ .

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Lemma 6.2Trivial bound

One has $μ (σ) = 0$ for all $σ \geq 1$ .

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Lemma 6.5Left of critical strip

One has $μ (σ) = 1 / 2 - σ$ for $σ \leq 0$ .

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Lemma 9.2Basic properties of

M (σ, A)

$M (σ, A)$ is convex in $σ$ .
For any $σ$ , $a (M (σ, 1 / a) - 1)$ is convex non-increasing in $a$ .
$M (σ, A) = 1$ for all $A \geq 0$ and $σ \geq 1$ .
$M (σ, A) \geq 1$ for all $A \geq 0$ and $1 / 2 \leq σ \leq 1$ .
$M (σ, 0) = 1$ for all $σ$ .
$M (1 - σ, A) = M (1 - σ, A) + (1 / 2 - σ) A$ for all $σ \in R$ and $A \geq 0$ .
For any $σ$ , $a (M (σ, 1 / a) - 1)$ converges to $μ (σ)$ as $a \to 0$ . In particular (by previous properties), $M (σ, A) \leq A μ (σ) + 1$ for all $σ \geq 0$ and $A \geq 0$ , and also $M (σ, A) \leq M (σ, A_{0}) + μ (σ) (A - A_{0})$ for $σ \geq 0$ and $A \geq A_{0} \geq 0$ .

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Definition 9.1Zeta moment exponents

For fixed $σ \in R$ and $A \geq 0$ , we define $M (σ, A)$ to be the least (fixed) exponent for which the bound

\int_{T}^{2 T} | ζ (σ + i t) |^{A} d t ≪ T^{M (σ, A) + o (1)}

holds for all unbounded $T > 1$ .

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Definition 18.2Zero free region

A zero-free region of the Riemann zeta function is a set $D \subset C$ for which $ζ (s) \neq 0$ for all $s \in D$ .

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Theorem 18.8Chudakov zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{1}{(\log t)^{3 / 4 + o (1)}}

for $t$ sufficiently large.

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Theorem 18.6Classical zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{A}{\log t} .

for an absolute constant $A > 0$ and $t$ sufficiently large.

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Theorem 18.7Littlewood zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{A \log \log t}{\log t}

for an absolute constant $A > 0$ and $t$ sufficiently large.

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Theorem 18.9Korobov-Vinogradov zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{A}{(\log t)^{2 / 3} (\log \log t)^{1 / 3}}

for an absolute constant $A > 0$ and $t$ sufficiently large.

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7	8	9	10	11	12	13	14	15	16	17	18	19	20
45	57	69	81	93	107	121	134	149	163	177	193	207	223

7	8	9	10	11	12	13	14	15	16	17	18	19	20
45	57	69	81	93	107	121	134	149	163	177	193	207	223

7	8	9	10	11	12	13	14	15	16	17	18	19	20
45	57	69	81	93	107	121	134	149	163	177	193	207	223