9 Moment growth for the zeta function

Definition 9.1 Zeta 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 $T > 1$ .

Such moments may be interpreted as the “average" order of the Riemann zeta function. It is not difficult to show that $M (σ, A)$ is a well-defined (fixed) real number. A non-asymptotic definition is that it is the least constant such that for every $ε > 0$ there exists $C > 0$ such that

\int_{T}^{2 T} | ζ (σ + i t) |^{A} d t \leq C T^{M (σ, A) + ε}

holds for all $T \geq C$ .

Lemma 9.2 Basic 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 $A \geq A_{0} \geq 0$ .

Proof ▶

The claim (i) follows from the Phragmen-Lindelöf principle. The claim (ii) follows from Hölder. The claim (iii) follows from standard upper and lower bounds on $ζ (σ + i t)$ for $σ \geq 1$ . The claim (iv) follows from (i)-(iii), and (v) is trivial. The claim (vi) follows easily from the functional equation.

For (vii), the bound $M (σ, A) \leq A μ (σ) + 1$ is trivial, which implies that

lim_{a \to 0} a (M (σ, 1 / a) - 1) \leq μ (σ) .

Suppose for contradiction that

lim_{a \to 0} a (M (σ, 1 / a) - 1) < μ (σ),

thus there is $δ > 0$ such that

M (σ, A) \leq A (μ (σ) - δ) + 1

for all $A \geq 0$ . By convexity, this gives

M (σ + ε, A) \leq A (μ (σ) - δ / 2) + 1

for all sufficiently small $ε$ , and then by the Cauchy integral formula and Hölder’s inequality we can conclude that

| ζ (σ + ε / 2 + i t) | ≪ | t |^{μ (σ) - δ / 2 + O (1 / A) + o (1)}

for unbounded $| t |$ , leading to

μ (σ + ε / 2) \leq μ (σ) - δ / 2 + O (1 / A) .

Sending $A$ to infinity and $ε$ to zero, we obtain a contradiction.

Corollary 9.3 Relationship 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.

Note from Lemma 9.2 that we always have the lower bound $M (σ, A) \geq 1 + max (1 / 2 - σ, 0) A$ . Thus there are not expected to be any further lower bound results for $M (σ, A)$ , and we focus now on upper bounds. Compared to the pointwise estimates $μ (σ)$ of $ζ (σ + i t)$ , which are currently open for all $0 < σ < 1$ , more are known about moment estimates $M (σ, A)$ . In particular,

Lemma 9.4

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

Proof ▶

Follows from Hölder’s inequality and the standard estimates

\int_{T}^{2 T} | ζ (1 / 2 + i t) |^{2} d t = T^{1 + o (1)}

and

\int_{T}^{2 T} | ζ (1 / 2 + i t) |^{4} d t = T^{1 + o (1)}

for any unbounded $T > 1$ , due to [ 67 ] and [ 68 ] respectively.

From Lemma 9.2 and Lemma 9.4 we may restrict attention to the region $1 / 2 \leq σ \leq 1$ and $A \geq 4$ .

9.1 Relationship to zeta large value estimates

We can relate $M (σ, A)$ to ${LV}_{ζ} (σ, τ)$ :

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 $τ \geq 2$ .

Proof ▶

We first show the lower bound, or equivalently that

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

whenever $τ \geq 2$ and $σ \geq 1 / 2$ . Accordingly, let $N$ be unbounded, $T = N^{τ + o (1)}$ , $I \subset [N, 2 N]$ , and $W$ be a $1$ -separated subset of $[T, 2 T]$ such that

| \sum_{n \in I} n^{- i t} | ≫ N^{σ + o (1)}

for $t \in W$ . By standard Fourier analysis (or by Perron’s formula and contour shifting), this gives

\int_{T / 2}^{3 T} | ζ (σ_{0} + i t^{'}) | \frac{d t}{1 + | t^{'} - t |} ≫ N^{σ - σ_{0} + o (1)}

and hence by Hölder

\int_{T / 2}^{3 T} | ζ (σ_{0} + i t^{'}) |^{A} \frac{d t^{'}}{1 + | t^{'} - t |} ≫ N^{A (σ - σ_{0}) + o (1)}

so on summing in $r$

\int_{T / 2}^{3 T} | ζ (σ_{0} + i t^{'}) |^{A} d t^{'} ≫ | W | N^{A (σ - σ_{0}) + o (1)} .

By Definition 9.1, the left-hand side is $≪ T^{M (σ_{0}, A) + o (1)}$ . Since $T = N^{α + o (1)}$ , we obtain

| W | ≪ N^{τ M (σ_{0}, A) - A (σ - σ_{0})},

giving the claim.

For the converse bound, let $M$ be the right-hand side of 1. From Lemma 8.9 we have $M \geq 1$ . By [ 110 , § 8.1 ] it will suffice to show that for any $V > 0$ and any $1$ -separated $W \subset [T, 2 T]$ with

| ζ (σ_{0} + i t) | \geq V

for all $t \in W$ , one has

| W | ≪ T^{M + o (1)} V^{- A} .

The claim is clear if $V \geq T^{C}$ or $V \leq T^{C}$ for some sufficiently large $C$ , so we may assume that $V = T^{O (1)}$ . We also clearly can assume $| W | \geq 1$ . Using the Riemann–Siegel formula [ 110 , Theorem 4.1 ] and dyadic decomposition, we have either

| \sum_{n \in I} \frac{1}{n^{σ_{0} + i t}} ≫ T^{- o (1)} V

T^{1 / 2 - σ_{0}} | \sum_{n \in I} \frac{1}{n^{1 - σ_{0} - i t}} | ≫ T^{- o (1)} V

for some $I \subset [N, 2 N]$ and $1 \leq N ≪ T^{1 / 2}$ , and all $t \in W$ . In either case, we can perform summation by parts and conclude that

| \sum_{n \in I^{'}} n^{- i t} | ≫ T^{- o (1)} V N^{σ_{0}}

| \sum_{n \in I^{'}} n^{- i t} | ≫ T^{- o (1)} V N^{1 - σ_{0}} T^{σ_{0} - 1 / 2}

for some $I^{'}$ in $[N, 2 N]$ and all $t \in W$ . As $σ_{0} \geq 1 / 2$ , the letter hypothesis is stronger than the former, so we may assume the former. If $N = T^{o (1)}$ then this would imply that $V ≪ T^{o (1)}$ , and we would be done from the trivial bound $R ≪ T$ since $M \geq 1$ . Hence, after passing to a subsequence, we can assume that $N = T^{1 / τ + o (1)}$ for some $2 < τ < \infty$ . We can also assume that $V = N^{σ - σ_{0} + o (1)}$ for some $σ \in R$ . If $σ \leq σ_{0}$ then $V ≪ T^{o (1)}$ and we are done as before, so we may assume $σ > σ_{0}$ ; in particular, $σ \geq 1 / 2$ . From Lemma 8.2 we have

| W | ≪ N^{{LV}_{ζ} (σ, τ) + o (1)}

and hence by definition of $M$

| W | ≪ N^{M τ - A (σ - σ_{0}) + o (1)} = T^{M + o (1)} V^{- A}

as required.

Corollary 9.6 Fourth moment bound

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

Proof ▶

Apply Lemma 9.5 with $σ_{0} = 1 / 2$ and $A = 4$ , using Lemma 9.2(iv).

We have an important twelfth moment estimate of Heath-Brown:

Theorem 9.7 Heath-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 $1 / 2 \leq σ \leq 1$ .

Proof ▶

From Lemma 8.11 with the exponent pair $(1 / 2, 1 / 2)$ from Lemma 5.10 we have

{LV}_{ζ} (σ, τ) \leq min (τ - 6 (σ - 1 / 2), 2 τ - 12 (σ - 1 / 2)) .

If $2 τ - 12 (σ - 1 / 2) \geq 0$ , the claim is immediate; if instead $2 τ - 12 (σ - 1 / 2) < 0$ , use Lemma 8.4.

We also have a variant bound, which is slightly better when $τ$ is close to $6 (σ - 1 / 2)$ :

Theorem 9.8 Auxiliary 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)) .

Proof ▶

Let $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ be a zeta large value pattern with $N$ , $V = N^{σ + o (1)}$ , $T = N^{τ + o (1)}$ and $W = N^{{LV}_{ζ} (σ, τ) + o (1)}$ . Our task is to show that

| W | ≪ T^{o (1)} (T (N^{- 1 / 2} V)^{- 6} + T^{5} (N^{- 1 / 2} V)^{- 32}) .

Write $a (n) = 1_{I} (n)$ . By a Fourier analytic expansion we can bound

N^{- 1 / 2} | \sum_{n \in I} n^{- i t} | ≪ T^{o (1)} \int_{T / 4}^{3 T} | ζ (1 / 2 + i t_{1}) | \frac{d t_{1}}{1 + | t_{1} - t |} + N^{- ε}

for some fixed $ε > 0$ and all $t \in W$ , hence

\int_{T / 4}^{3 T} | ζ (1 / 2 + i t_{1}) | \frac{d t}{1 + | t_{1} - t |} ≫ T^{- o (1)} N^{- 1 / 2} V .

In particular, we can truncate to large values of $ζ (1 / 2 + i t_{1})$ , in the sense that

\int_{T / 4}^{3 T} | ζ (1 / 2 + i t_{1}) | 1_{| ζ (1 / 2 + i t_{1}) | \geq T^{- o (1)} N^{- 1 / 2} V} \frac{d t_{1}}{1 + | t_{1} - t |} ≫ T^{- o (1)} N^{- 1 / 2} V .

Summing in $t$ and using the $1$ -separation to bound the sum of $1 / (1 + | t_{1} - t |)$ by $T^{o (1)}$ , we conclude that

\int_{T / 4}^{3 T} | ζ (1 / 2 + i t_{1}) | 1_{| ζ (1 / 2 + i t_{1}) | \geq T^{- o (1)} N^{- 1 / 2} V} d t_{1} ≫ T^{- o (1)} R N^{- 1 / 2} V .

Hence by dyadic pigeonholing we have

V^{'} \int_{T / 4}^{3 T} | ζ (1 / 2 + i t_{1}) | 1_{| ζ (1 / 2 + i t_{1}) | ≍ V^{'}} d t ≫ T^{- o (1)} R N^{- 1 / 2} V

for some $V^{'} \geq T^{- o (1)} N^{- 1 / 2} V$ , and thus

| ζ (1 / 2 + i t^{'}) | ≍ V^{'}

for all $t^{'}$ in some $1$ -separated subset $W^{'}$ of $[T / 4, 3 T]$ with

| W^{'} | ≫ T^{- o (1)} | W | N^{- 1 / 2} V / V^{'} .

Applying [ 75 , Theorem 2 ] (treating different cases using the bounds [ 75 , (7), (8), (9) ] ), we have the bound

| W^{'} | ≪ T^{o (1)} (T (V^{'})^{- 6} + T^{5} (V^{'})^{- 32})

and thus

| W | ≪ T^{o (1)} (T (N^{- 1 / 2} V)^{- 1} (V^{'})^{- 5} + T^{5} (N^{- 1 / 2} V)^{- 1} (V^{'})^{- 31})

and the claim now follows from the lower bound on $V^{'}$ .

9.2 Known moment growth bounds

Lemma 9.9 Ivic’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 $A = 12$ .

Proof ▶

Theorem 9.10 Moment 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} .

In particular, we have

\begin{aligned} M (1 / 2, 13) & \leq 2.1340, & M (1 / 2, 14) & \leq 2.2720, & M (1 / 2, 15) & \leq 2.4137, \\ M (1 / 2, 16) & \leq 2.5570, & M (1 / 2, 17) & \leq 2.7016, & M (1 / 2, 18) & \leq 2.8466 . \end{aligned}

9.3 Large values of $ζ$ moments

It is also of interest to control large values of the moments.

Definition 9.11 Mixed 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 $T$ .

Lemma 9.12 Mixed 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 $σ \leq h τ$ .

Proof ▶

Corollary 9.13 Mixed 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} .

Proof ▶

From Lemma 8.11 with the exponent pair $(k, ℓ)$ we have

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

In particular, for $σ - 1 / 2 \leq h τ$ one has

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

and for $σ - 1 / 2 \geq h τ$ one has

\frac{k + ℓ}{k} τ - \frac{2 (1 + 2 k + 2 ℓ)}{k} (σ - 1 / 2) .

The claim then follows from Lemma 9.12.

Corollary 9.14 Specific mixed moments

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

Proof ▶

Apply Corollary 9.13 with the exponent pair $(4 / 18, 11 / 18) = B A B A (1 / 6, 2 / 3)$ from Corollary 5.11. □

Lemma 9.15 Large 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} .

Lemma 9.16 Zero 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}) .

It is remarked in [ 19 ] that this proposition could lead to some improvements in current zero density estimate bounds.

Lemma 9.17 Chen-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) .

In [ 26 ] this lemma is applied with $(q_{0}, A_{0}) = (6, 1)$ and $(q_{1}, A_{1}) = (19, 3)$ with $h = 2 / 13$ , which follows from Corollary 9.13 applied to the exponent pair $(2 / 7, 4 / 7) = B A (1 / 6, 2 / 3)$ from Corollary 5.11.

9 Moment growth for the zeta function

9.1 Relationship to zeta large value estimates

9.2 Known moment growth bounds

9.3 Large values of ζ moments

9.3 Large values of $ζ$ moments