8 Large value theorems for zeta partial sums

Now we study a variant of the exponent $LV (σ, τ)$ , specialized to the Riemann zeta function.

Definition 8.1 Large 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 $| W | ≪ N^{ρ + o (1)}$ .

Implemented at large_values.py as:
Large_Value_Estimate

We will primarily be interested in the regime $τ \geq 2$ (as this is the region connected to the Riemann-Siegel formula for $ζ (σ + i t)$ ), but for sake of completeness we develop the theory for the entire range $τ \geq 0$ . (The range $0 \leq τ \leq 1$ can be worked out exactly by existing tools, while the region $1 < τ < 2$ can be reflected to the region $2 < τ < \infty$ by Poisson summation.)

As usual, we have a non-asymptotic formulation of ${LV}_{ζ} (σ, τ)$ :

Lemma 8.2 Asymptotic 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

$| W | \leq C N^{ρ + ε} .$

The proof of Lemma 8.2 proceeds as in previous sections and is omitted.

Lemma 8.3 Basic 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

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

Implemented at zeta_large_values.py as:
get_trivial_zlv()

We note that in practice, bounds for ${LV}_{ζ} (σ^{'}, τ) + σ^{'}$ are monotone decreasing ¹ in $σ^{'}$ , so the reflection property in Lemma 8.3(iv) morally simplifies ² to

{LV}_{ζ} (\frac{1}{2} + \frac{1}{τ - 1} (σ - \frac{1}{2}), \frac{τ}{τ - 1}) = \frac{1}{τ - 1} {LV}_{ζ} (σ, τ) .

TODO: implement a python method for reflection

Proof ▶

The claims (i), (ii) are obvious. The claim (iii) is clear by setting $a_{n} = 1_{I}$ in Definition 7.2.

Now we turn to (iv). By symmetry it suffices to prove the upper bound. Actually it suffices to just show

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

as this easily implies the general upper bound.

Let $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ be a zeta large value pattern with $N$ unbounded, $T = N^{\frac{τ}{τ - 1} + o (1)}$ , and $V = N^{\frac{1}{2} + \frac{1}{τ - 1} (σ - \frac{1}{2}) + o (1)}$ . By definition, it suffices to show the bound

| W | ≪ N^{\frac{1}{τ - 1} ({LV}_{ζ} (σ^{'}, τ) + σ^{'} - σ) + o (1)} .

for some $σ \leq σ^{'} \leq 1$ . By definition, $a_{n} = 1_{I} (n)$ . By a Fourier expansion of $(n / N)^{1 / 2}$ in $\log n$ , we can bound

| \sum_{n \in I} n^{- i t_{r}} | ≪_{A} N^{1 / 2} \int_{R} | \sum_{n \in I} n^{- 1 / 2 - i t} | (1 + | t - t_{r} |)^{- A} d t

and hence by the pigeonhole principle, we can find $t^{'} = t + O (N^{o (1)})$ for each $t \in W$ such that

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

for $t \in W$ . By refining $W$ by $N^{o (1)}$ if necessary, we may assume that the $t^{'}$ are $1$ -separated.

Now we use the approximate functional equation

ζ (1 / 2 + i t^{'}) = \sum_{n \leq x} n^{- 1 / 2 - i t^{'}} + χ (1 / 2 + i t^{'}) \sum_{m \leq t^{'} / 2 π x} m^{- 1 / 2 + i t^{'}} + O (N^{- 1 / 2}) + O ((T / N)^{- 1 / 2})

for $x \sim N$ ; see [ 110 , Theorem 4.1 ] . Applying this to the two endpoints of $I$ and subtracting, we conclude that

\sum_{n \in I} n^{- 1 / 2 - i t^{'}} = χ (1 / 2 + i t^{'}) \sum_{m \in J_{t^{'}}} m^{- 1 / 2 + i t^{'}} + O (N^{- 1 / 2}) + O ((T / N)^{- 1 / 2})

where $J_{t^{'}} := {m : t^{'} / 2 π m \in I}$ . Since $χ (1 / 2 + i t^{'})$ has magnitude one, we conclude that

| \sum_{m \in J_{r}} m^{- 1 / 2 - i t^{'}} | ≫ N^{- 1 / 2 - o (1)} V .

Writing $M := T / N = N^{\frac{1}{τ - 1} + o (1)}$ , we see that $J_{r} \subset [M / 10, 10 M]$ and

| \sum_{m \in J_{r}} (M / m)^{1 / 2} m^{- i t^{'}} | ≫ M^{1 / 2} N^{- 1 / 2 - o (1)} V = M^{σ + o (1)} .

Performing a Fourier expansion of $(M / m)^{1 / 2} 1_{J_{r}} (m)$ (smoothed out at scale $O (1)$ ) in $\log m$ , we can bound

| \sum_{m \in J_{r}} (M / m)^{1 / 2} m^{- i t^{'}} | ≪ \int_{T / 10}^{10 T} | \sum_{m \in [M / 10, 10 M]} m^{- i t_{1}} | (1 + | t_{1} - t^{'} |)^{- 1} d t_{1} + T^{- 10}

and hence

\int_{T / 10}^{10 T} | \sum_{m \in [M / 10, 10 M]} m^{- i t_{1}} | (1 + | t_{1} - t^{'} |)^{- 1} d t_{1} ≫ M^{σ + o (1)} .

If we let $E$ denote the set of $t_{1} \in [T / 10, 10 T]$ for which $| \sum_{m \in [M / 10, 10 M]} m^{- i t_{1}} | \geq M^{σ - o (1)}$ for a suitably chosen $o (1)$ error, then we have

\int_{E} | \sum_{m \in [M / 10, 10 M]} m^{- i t_{1}} | (1 + | t_{1} - t^{'} |)^{- 1} d t ≫ M^{σ + o (1)} .

Summing in $t^{'}$ , we obtain

\int_{E} | \sum_{m \in [M / 10, 10 M]} m^{- i t_{1}} | d t_{1} ≫ M^{σ + o (1)} R

and so by dyadic pigeonholing we can find $M^{σ - o (1)} ≪ V^{″} ≪ M$ and a $1$ -separated subset $W^{″}$ of $E$ such that

| \sum_{m \in [M / 10, 10 M]} m^{- i t^{″}} | d t ≍ V^{″}

for all $t^{″} \in W^{″}$ , and

V^{″} | W^{″} | ≫ M^{σ + o (1)} | W | .

By passing to a subsequence we may assume that $V^{″} = M^{σ^{'} + o (1)}$ for some $σ \leq σ^{'} \leq 1$ . Partitioning $[M / 10, 10 M]$ into a bounded number of intervals each of which lies in a dyadic range $[M^{'}, 2 M^{'}]$ for some $M^{'} ≍ M$ , and using Definition 8.1, we have

| W^{″} | ≪ M^{{LV}_{ζ} (σ^{'}, τ) + o (1)}

and 2 follows.

Note in comparison with $LV (σ, τ)$ , that ${LV}_{ζ} (σ, τ)$ can be $- \infty$ , and is indeed conjectured to do so whenever $σ > 1 / 2$ and $τ \geq 1$ . Indeed:

Lemma 8.4 Characterization 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 $| t | = N^{τ + o (1)}$ .

Proof ▶

Clearly (i) implies (ii). If (iii) holds, then in any zeta large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N$ unbounded and $V = N^{σ + o (1)}$ , $W$ is necessarily empty, giving (i). Conversely, if (i) fails, then there must be $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N$ unbounded and $V = N^{σ + o (1)}$ with $W$ non-empty, contradicting (ii).

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.

Proof ▶

Suppose one has data $N, I$ obeying the hypotheses of Lemma 8.4(iii), then by 2 (with $α = 1 / τ$ ) one has

\sum_{n \in I} n^{- i t} ≪ | t |^{β (1 / τ) + o (1)} = N^{τ β (1 / τ) + o (1)}

and the claim follows from Lemma 8.4.

Corollary 8.6

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

Proof ▶

From Definition 6.1 one has

ζ (σ_{0} + i t) ≪ | t |^{μ (σ_{0}) + o (1)}

for unbounded $t$ . By standard arguments (see [ 110 , (8.13) ] ), this implies that

\sum_{n \in I} \frac{1}{n^{σ_{0} + i t}} ≪ | t |^{μ (σ_{0}) + o (1)}

for unbounded $N$ , if $I \subset [N, 2 N]$ and $| t | = N^{τ + o (1)}$ . By partial summation this gives

\sum_{n \in I} n^{- i t} ≪ N^{σ_{0}} | t |^{μ (σ_{0}) + o (1)} = N^{σ_{0} + τ μ (σ_{0}) + o (1)} .

The claim now follows from Lemma 8.4.

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 $σ > k τ + ℓ - k$ .

Proof ▶

Corollary 8.8

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

Proof ▶

Apply Corollary 8.6 with $σ_{0} = 1 / 2$ , so that $μ (σ_{0})$ vanishes from the Lindelof hypothesis.

For completeness, we now work out the values of ${LV}_{ζ} (σ, τ)$ in the remaining cases not covered by the above corollary.

Lemma 8.9 Value at

σ = 1 / 2

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

Proof ▶

The upper bound ${LV}_{ζ} (1 / 2, τ) \leq τ$ follows from Lemma 8.3(ii), so it suffices to prove the lower bound. Accordingly, let $N$ be unbounded, let $T = C N$ for a large fixed constant $C$ , and set $I := [N, 2 N]$ . In the case $σ = 1$ , we see from the $L^{2}$ mean value theorem (Lemma 3.1) that the expression $\sum_{n \in I} n^{- i t}$ has an $L^{2}$ mean of $≍ N^{1 / 2}$ for $t \in [T, 2 T]$ ; on other hand, from 8 we also have an $L^{\infty}$ norm of $O (N^{1 / 2 + o (1)})$ . We conclude that $| \sum_{n \in I} n^{- i t} | ≫ N^{1 / 2 + o (1)}$ for $t$ in a subset of $[T, 2 T]$ of measure $T^{1 - o (1)}$ , and hence on a $1$ -separated subset of cardinality $≫ T^{1 - o (1)}$ . This gives the claim $LV (1 / 2, 1) \geq 1$ .

Next, we establish the $τ \geq 2$ case. Let $N$ be unbounded, set $T := N^{τ}$ , and set $I := [N, 2 N]$ . From Lemma 3.1 we see that the $L^{2}$ mean of $\sum_{n \in I} n^{- i t}$ is $≍ N^{1 / 2}$ . Also, by squaring this Dirichlet series and applying Lemma 3.1 again we see that the $L^{4}$ mean is $O (N^{1 / 2 + o (1)})$ . We may now argue as before to give the desired claim $LV (1 / 2, τ) \geq τ$ .

Finally we need to handle the case $1 < τ < 2$ . By Lemma 8.3(iv) with $σ = 1 / 2$ we have

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

By the $τ \geq 2$ case, the left-hand side is at least $τ / (τ - 1)$ , thus

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

On the other hand, from Theorem 7.9 and Lemma 8.3(iii) we have

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

We conclude that the supremum is in fact attained asymptotically at $σ^{'} = 1 / 2$ , in the sense that

\underset{σ^{'} \to 1 / 2^{+}}{lim sup} {LV}_{ζ} (σ^{'}, τ) + σ^{'} - 1 / 2 \geq τ .

By the monotonicity of ${LV}_{ζ}$ in $σ$ , this implies that ${LV}_{ζ} (1 / 2, τ) \geq τ$ , as required.

Lemma 8.10 Value 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 $σ \leq 1 - τ$ .

Proof ▶

The first claim follows from Corollary 8.5 and Lemma 4.4. For the second claim, it suffices by Lemma 8.3(ii) to establish the lower bound ${LV}_{ζ} (σ, τ) \geq τ$ . But this is clear from 5.

One can use exponent pairs to control ${LV}_{ζ} (σ, τ)$ :

Lemma 8.11 From 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)) .

By applying this lemma to the exponent pairs in Corollary 5.11, one recovers the bounds in [ 110 , Corollary 8.1, 8.2 ] (up to epsilon losses in the exponents).

A useful connection between large values estimates and large values estimates for the zeta function is the following strengthening of Theorem 7.10.

Lemma 8.12 Halász–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^{'})) .

Note from Lemma 8.5 one could also impose the restriction $σ^{'} \leq τ^{'} β (1 / τ^{'})$ in the supremum if desired, at which point one recovers Theorem 7.10. Similarly, from Corollary 8.6 one could also impose the restriction $σ^{'} \leq σ_{0} + τ^{'} μ (σ_{0})$ for any fixed $1 / 2 \leq σ_{0} \leq 1$ .

Proof ▶

It suffices to show that

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

since the terms with $σ^{'} < 2 σ - 1$ are less than the left-hand side and can thus be dropped. We repeat the proof of Lemma 7.10. We can find a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N$ unbounded, $V = N^{σ + o (1)}$ , $T = N^{τ + o (1)}$ , and $| W | = N^{LV (σ, τ) + o (1)}$ , and we have

| W | V \leq N^{1 / 2} {| \sum_{t, t^{'} \in W} c_{t} \overset{―}{c_{t^{'}}} \sum_{n \in [N, 2 N]} n^{i (t - t^{'})} |}^{1 / 2}

for some $1$ -bounded $c_{t}$ , and hence by the triangle inequality

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

which we rearrange as

| W | \leq N^{1 - 2 σ + o (1)} sup_{t^{'}} \sum_{t \in W} | \sum_{n \in [N, 2 N]} n^{i (t - t^{'})} | .

As in the proof of Lemma 7.10, the contribution of the case $| t - t^{'} | \leq N^{1 - ε}$ to the right-hand side is $N^{2 - 2 σ + o (1)}$ , so we can restrict attention to the case $| t - t^{'} | \geq N^{1 - o (1)}$ . By a dyadic decomposition and the pigeonhole principle, we may then assume that

| W | \leq N^{1 - 2 σ + o (1)} \sum_{t \in W : | t - t^{'} | ≍ T^{'}} | \sum_{n \in [N, 2 N]} n^{i (t_{r} - t_{r^{'}})} |

for some $N^{1 - o (1)} ≪ T^{'} ≪ T$ and some $r^{'}$ ; by passing to a subsequence we may assume that $T^{'} = N^{τ^{'} + o (1)}$ for some $1 \leq τ^{'} \leq τ$ . By further dyadic decomposition, we may also assume that $| \sum_{n \in [N, 2 N]} n^{i (t - t^{'})} | ≍ N^{σ^{'} + o (1)}$ for some $σ^{'} \leq 1$ ; the cardinality of the sum is then bounded both by $| W |$ and by $N^{{LV}_{ζ} (σ^{'}, τ^{'}) + o (1)}$ , hence

| W | \leq N^{1 - 2 σ + σ^{'} + min (LV (σ, τ), {LV}_{ζ} (σ^{'}, τ^{'})) + o (1)} .

The case $σ^{'} < 1 / 2$ is dominated by that of $σ^{'} = 1 / 2$ . The claim now follows.

Corollary 8.13 Converting 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 $τ \leq \frac{2 σ - 1 - σ^{'}}{μ (σ^{'})}$ .

Proof ▶

By Lemma 7.7 it suffices to verify the claim for $τ < \frac{2 σ - 1 - σ^{'}}{μ (σ^{'})}$ . The claim now follows from Lemma 8.12 and Corollary 8.6.

Theorem 8.14 Halász-Turán large values theorem

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

Proof ▶

Immediate from Corollary 8.13, since $μ (1 / 2) = 0$ in this case.

Theorem 8.15 First 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^{'} .

Proof ▶

We set $θ$ to equal

(3 σ - 2) / (2 σ - 1) for 1 / 2 < σ \leq 2 / 3;

(9 σ - 6) / (4 σ - 1) for 2 / 3 \leq σ \leq 11 / 14;

(25 σ - 16) / (8 σ + 1) for 11 / 14 \leq σ \leq 13 / 15;

(65 σ - 40) / (16 σ + 9) for 13 / 15 \leq σ \leq 57 / 62;

(12 σ - 7) / (2 σ + 3) for 57 / 62 \leq σ < 1,

and then from the bounds $μ (1 / 2) \leq 1 / 6$ , $μ (5 / 7) \leq 1 / 14$ , $μ (5 / 6) \leq 1 / 30$ one can bound $μ (θ)$ by the quantity $c (θ)$ , defined to equal

1 / 2 - θ for θ \leq 0

(3 - 4 θ) / 6 for 0 \leq θ \leq 1 / 2

(7 - 8 θ) / 18 for 1 / 2 \leq θ \leq 5 / 7

(15 - 16 θ) / 50 for 5 / 7 \leq θ \leq 5 / 6

(1 - θ) / 5 for 5 / 6 \leq θ \leq 1.

By Corollary 8.13, we have $LV (σ^{'}, τ) \leq 2 - 2 σ^{'}$ for

τ \leq \frac{2 σ^{'} - 1 - θ}{c (θ)} .

The right-hand side can be computed to equal $f (σ) (σ^{'} - σ) + 2 - 2 σ^{'}$ , giving the claim.

Another typical application of the Halász-Montgomery inequality is

Lemma 8.16 Second 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}) .

Proof ▶

Write $ρ := LV (σ, τ)$ , and let $ε > 0$ be arbitrary. By Lemma 8.12, we may assume without loss of generality that

ρ \leq max (2 - 2 σ, 1 - 2 σ + σ^{'} + min (ρ, {LV}_{ζ} (σ^{'}, τ^{'}))) + ε

for some $1 / 2 \leq σ^{'} \leq 1$ and $1 \leq τ^{'} \leq τ$ . On the other hand, from Lemma 8.11 applied to the exponent pair $(2 / 7, 4 / 7)$ from Lemma 5.11, and bounding $τ^{'}$ by $τ$ , one has

{LV}_{ζ} (σ^{'}, τ^{'}) \leq max (τ - 6 (σ^{'} - 1 / 2), 3 τ - 19 (σ^{'} - 1 / 2))

and thus on taking convex combinations

min (ρ, {LV}_{ζ} (σ^{'}, τ^{'})) \leq max (\frac{5}{6} ρ + \frac{1}{6} τ - (σ^{'} - 1 / 2), \frac{18}{19} ρ + \frac{3}{19} τ - (σ^{'} - 1 / 2)),

hence $ρ$ is bounded by either $2 - 2 σ$ , $1 - 2 σ + \frac{5}{6} ρ + \frac{1}{6} τ + \frac{1}{2}$ , or $1 - 2 σ + \frac{18}{19} ρ + \frac{3}{19} τ + \frac{1}{2}$ . The claim then follows after solving for $ρ$ . □