On a conjecture of Montgomery and Soundararajan

Abstract

We establish lower bounds for all weighted even moments of primes up to X in intervals which are in agreement with a conjecture of Montgomery and Soundararajan. Our bounds hold unconditionally for an unbounded set of values of X, and hold for all X under the Riemann Hypothesis. We also deduce new unconditional $\Omega$-results for the classical prime counting function.

1 Introduction

The goal of this paper is to investigate [24, Conjecture 1]. Let

$$\begin{aligned} \mu _n:= {\left\{ \begin{array}{ll} \frac{(2m)!}{2^m m!} &{} \text { if } n=2m \text { for some } m\in {\mathbb {N}},\\ 0 &{}\text {otherwise} \end{array}\right. } \end{aligned}$$

(1.1)

be the n-th moment of the Gaussian.

Conjecture 1.1

(Montgomery, Soundararajan) Fix $\varepsilon >0$. For each fixed $n\in {\mathbb {N}}$ and uniformly for $\frac{(\log X)^{1+\varepsilon }}{X}\leqslant \delta \leqslant \frac{1}{X^{\varepsilon }}$,

$$\begin{aligned} \frac{1}{ X}\int _1^{X} \frac{(\psi (x+\delta X)-\psi (x)-\delta X)^n}{X^{\frac{n}{2}}} \mathrm{d}x = (\mu _n+o(1)) \big ( \delta \log (\delta ^{-1})\big )^{\frac{n}{2}}. \end{aligned}$$

(1.2)

In the range $X^{-1} (\log X)^{1+\varepsilon } \leqslant \delta \leqslant X^{-1+\frac{1}{n}}$, Montgomery and Soundararajan [24, Theorem 3] have shown that (1.2) follows from a strong form of the Hardy-Littlewood prime k-tuple conjecture. For applications on the distribution of gaps between primes, see for instance [2].

Currently, many results towards Conjecture 1.1 are known in the case $n=2$ (see the remarks following Theorem 1.2 below for a description of the work of Selberg, Goldston, Montgomery, and others on this topic), but little is known for higher moments. This is in contrast with the theory of moments of L-functions, in which we have lower and upper bounds of the correct order of magnitude for higher moments in several different families thanks to the work of Ramachandra [29], Rudnick and Soundararajan [31], Soundararajan [34], Harper [14], Radziwiłł-Soundararajan [28], and others.

In the current paper, we establish lower bounds for a weighted version of (1.2) for all even n, for values of $\delta$ that are relatively close to 1. In addition to being the first estimate on higher moments, we believe that our bounds are sharp up to a power-saving error term in $\delta$ (cf. [24, Theorem 3]). Prior to our work, the order of magnitude of the left hand side of (1.2) and some variants was known under RH for $n=2$ and in various ranges of $\delta$. However, the determination of the exact asymptotic size has been shown to be strongly related with deep simplicity and pair-correlation type estimates [1, 3, 5, 8, 13, 22, 23, 26].

The key idea which will allow us to circumvent the need to understand spacing statistics and Diophantine properties (for higher moments) of zeros of the zeta function is a positivity argument in the explicit formula. Such an argument in conjunction with Parseval’s identity has been successfully used in previous works on the variance (see e.g. [7]), however the novelty in the present paper is to avoid the need for Parseval’s identity (in particular for higher moments).

For any fixed $\kappa >0$, we define the class of test functions ${\mathcal {E}}_\kappa \subset {\mathcal {L}}^1({\mathbb {R}})$ to be the set of all differentiable¹ even $\eta :{\mathbb {R}} \rightarrow {\mathbb {R}}$ such that for all $t\in {\mathbb {R}}$,

$$\begin{aligned} \eta (t) , \eta '(t) \ll \mathrm{e}^{-\kappa |t|}, \end{aligned}$$

(1.3)

moreover ${{\widehat{\eta }}}(0)>0$ and for all $\xi \in {\mathbb {R}}$ we have that²

$$\begin{aligned} 0 \leqslant {{\widehat{\eta }}}(\xi ) \ll (|\xi |+1)^{-1} \log (|\xi |+2)^{-2-\kappa }. \end{aligned}$$

(1.4)

We consider the following weighted version of $x^{-\frac{1}{2}}(\psi (x+\delta x)-\psi (x)-\delta x)$. For $\eta \in {\mathcal {E}}_{\kappa }$ and $\delta < 2\kappa$, we define

$$\begin{aligned} \psi _{\eta } (x,\delta ):= \sum _{n\geqslant 1} \frac{\Lambda (n)}{n^{\frac{1}{2}}} \eta \Big (\delta ^{-1}\log \Big (\frac{n}{x}\Big )\Big ). \end{aligned}$$

Morally, this function counts prime powers in the interval $[x (1-O(\delta )),x(1+O(\delta ))]$, in which the weight $n^{-\frac{1}{2}}$ is equal to $x^{-\frac{1}{2}}(1+O(\delta ))$. The expected main term for $\psi _{\eta } (x,\delta )$ is given by

$$\begin{aligned} \int _{0}^{\infty } \frac{\eta (\delta ^{-1}\log (\frac{t}{x}))}{t^{\frac{1}{2}}} \mathrm{d}t = x^{\frac{1}{2}} \delta \int _{{\mathbb {R}}} \mathrm{e}^{\frac{\delta w}{2}} \eta (w) \mathrm{d}w , \end{aligned}$$

which we will denote by $x^{\frac{1}{2}} \delta {\mathcal {L}}_\eta (\frac{\delta }{2})$ (note that for $\delta <\kappa$, ${\mathcal {L}}_\eta (\frac{\delta }{2})={\mathcal {L}}_\eta (-\frac{\delta }{2})= {{\widehat{\eta }}}(0)+O(\delta )$). Subtracting this main term is equivalent to summing $\Lambda (n)-1$ instead of $\Lambda (n)$ (more precisely, it is equivalent to working with the measure $\mathrm{d}(\psi (t)-t)$). We also consider the set ${\mathcal {U}}$ of non-trivial even integrable functions $\Phi :{\mathbb {R}} \rightarrow {\mathbb {R}}$ such that $\Phi ,{{\widehat{\Phi }}}\geqslant 0$ (in particular, $\Phi (0)>0$). Finally, for $h:{\mathbb {R}} \rightarrow {\mathbb {R}}$ we define

$$\begin{aligned} \alpha (h):= \int _{{\mathbb {R}} } h(t)\mathrm{d}t; \qquad \beta (h):= \int _{{\mathbb {R}} } h(t) (\log | t|)\mathrm{d}t, \end{aligned}$$

(1.5)

whenever these integrals converge. Here is our main RH result on the n-th moment

$$\begin{aligned} M_n(X,\delta ;\eta ,\Phi ) :=\frac{1}{ (\log X) \int _0^{\infty }\Phi }\int _{1}^{\infty } \Phi \Big ( \frac{\log x}{\log X}\Big )\big (\psi _\eta (x,\delta )-x^{\frac{1}{2}}\delta {\mathcal {L}}_\eta \big (\tfrac{\delta }{2}\big )\big )^{n} \frac{\mathrm{d}x}{x}. \end{aligned}$$

Theorem 1.2

Assume RH, and let $0<\kappa < \frac{1}{2}$, $\eta \in {\mathcal {E}}_{\kappa }$, $\Phi \in {\mathcal {U}}$. For $n\in {\mathbb {N}}$, $X\in {\mathbb {R}}_{\geqslant 2}$, $\delta \in (0,\kappa )$, and in the range $n\leqslant \delta ^{-\frac{1}{2}}(\log (\delta ^{-1}+2))^{\frac{1}{2}}$, we have that

$$\begin{aligned} \begin{aligned} (-1)^n M_n(X,\delta ;\eta ,\Phi )&\geqslant \mu _n \delta ^{\frac{n}{2}}\big (\alpha ({{\widehat{\eta }}}^2) \log (\delta ^{-1})+\beta ({{\widehat{\eta }}}^2)\big )^{\frac{n}{2}}\\&\Big (1+O_{\kappa ,\eta }\Big (\frac{ n^2 \delta }{ \log (\delta ^{-1}+2)}\Big )\Big ) + O_{\Phi }\Big (\delta \frac{(K_{\eta } \log (\delta ^{-1}+2))^{n }}{\log X}\Big ), \end{aligned} \end{aligned}$$

(1.6)

where the implied constants and $K_\eta >0$ are independent of n, X and $\delta$.

Remark 1.1

• (1) For $n=2$ and in the range $X^{-c({\eta ,\Phi })} \leqslant \delta \leqslant 1$, (1.6) implies a lower bound with the predicted main term as well as a secondary term conjectured in the work of Montgomery and Soundararajan [23, (2)]. Here, $c({\eta ,\Phi })>0$ is a constant. Variants of this particular case (with various weights and measures) have attracted a lot of attention since Selberg’s foundational work [33]. This includes Goldston and Montgomery’s RH upper bound [8] in the whole range $0<\delta \leqslant 1$, Saffari and Vaughan’s unconditional upper bound [32] in the range $x^{-\frac{5}{6}+\varepsilon }\leqslant \delta \leqslant 1$, Goldston’s GRH lower bound [4, 6] in the range $x^{-1}\leqslant \delta \leqslant x^{-\frac{3}{4}}$ (unconditional for $x^{-1}\leqslant \delta \leqslant x^{-1} (\log x)^A$), its generalization by Özlük [27] and Goldston and Yildirim [9, 10] to a fixed arithmetic progression, Zaccagnini’s unconditional upper bound [36, 37] in the range $x^{-\frac{5}{6}-\varepsilon } \leqslant \delta \leqslant 1$ (building on the work of Huxley [17] and Heath-Brown [16]), and others.

• (2) For $n=2m$ with $m\geqslant 2$ and in the interval $(\log X)^{-\frac{1}{m-1} +o(1)}\leqslant \delta \ll 1$, we obtain a lower bound which is in agreement with Conjecture 1.1.

• (3) Goldston and Yildirim [11, 12] have computed the first three moments of a related quantity involving a major arcs approximation of $\Lambda (n)$, and deduced that in the range $X\leqslant x \leqslant 2X$, $X^{-1} (\log X)^{14} \ll \delta \leqslant X^{-\frac{6}{7}-\varepsilon }$ and under GRH, $\psi (x+\delta X)- \psi (x) - \delta X =\Omega _{\pm }((\delta x \log x)^{\frac{1}{2}})$.

• (4) In the function field case, estimates for the variance of $\Lambda (n)$ and more general arithmetic sequences have been established by Keating and Rudnick [19, 20] and Rodgers [30]. Moreover, Hast and Matei [15] have given a geometric interpretation for the higher moments.

We now rephrase Theorem 1.2 and state our unconditional results.

Corollary 1.3

Let $0<\kappa <\frac{1}{2}$, $\eta \in {\mathcal {E}}_{\kappa }$, and $\Phi \in {\mathcal {U}}$. Let moreover $f : {\mathbb {R}}_{\geqslant 0} \rightarrow (0,\frac{1}{2}]$ be any function such that $\lim _{x \rightarrow \infty } f(x) = 0$, and let $\delta \in (0,1)$, $m \in {\mathbb {N}}$ and $X \in {\mathbb {R}}_{\geqslant 2}$ be such that either $m=1$ and $\delta \in (X^{-f(X)},f(X)]$, or $2\leqslant m\leqslant \min (\delta ^{-\frac{1}{2}} (\log (\delta ^{-1}+2))^{\frac{1}{2}}f(X)^{\frac{1}{2}},\log \log X)$ and $\delta \in ((\log X)^{-\frac{1}{m-1}} (\log \log X)^4,f(X)].$ Then under RH we have that

$$\begin{aligned} M_{2m}(X,\delta ;\eta ,\Phi ) \geqslant \mu _{2m} \delta ^{m}\big (\alpha ({{\widehat{\eta }}}^2) \log (\delta ^{-1})+\beta ({{\widehat{\eta }}}^2)\big )^{m}\Big (1+O\Big (f(X)+ \frac{1}{(\log (\delta ^{-1}))^2}\Big )\Big ).\nonumber \\ \end{aligned}$$

(1.7)

Unconditionally, there exists a sequence $\{X_j\}_{j\geqslant 1}$ tending to infinity such that whenever $X=X_j$, (1.7) holds with $m=1$ and $\delta \in (X^{-f(X)},f(X)]$. The same statement holds in the range $2\leqslant m\leqslant \min (\delta ^{-\frac{1}{2}},\log \log X)$ and $\delta \in ((\log X)^{-\frac{1}{ m-1} }(\log \log X)^4,f(X)]$.

We now state our unconditional $\Omega$-results for the usual prime counting function in short intervals $\psi (x+\delta x)-\psi (x) -\delta x$. Note that this quantity has standard deviation of order $(\delta x \log (\delta ^{-1}+2))^{\frac{1}{2}}$. We will show that $\psi (x+\delta x)-\psi (x) -\delta x$ can be larger than an unbounded constant times this.

Corollary 1.4

Let $\varepsilon >0$ be small enough. There exists a sequence $\{(x_j,\delta _j)\}_{j\geqslant 1}$ with $\delta _j\in \big [\varepsilon \frac{(\log _3 x_j)^{\frac{9}{2}}}{(\log x_j)^{2} (\log _2 x_j)^{\frac{5}{2}}},2\frac{(\log _3 x_j)^3}{(\log _2 x_j)^{2}}\big ]$, $\lim _{j\rightarrow \infty } x_j=\infty$, and such that

$$\begin{aligned} \big |\psi (x_j+\delta _j x_j)-\psi (x_j) - \delta _j x_j \big | \gg \delta _j^{-\frac{1}{4}}(\log (\delta _j^{-1}+2))^{\frac{1}{4}} \cdot \big (\delta _j x_j \log (\delta _j^{-1}+2)\big )^{\frac{1}{2}}. \end{aligned}$$

If instead we require that $\delta _j\in \big [(\log x_j)^{-\frac{7}{2}-\frac{3}{2M}},(\log x_j)^{-\frac{1}{M+1}}\big ]$ for some large fixed $M\in {\mathbb {Z}}_{\geqslant 2}$, then we can choose the sequence $\{(x_j,\delta _j)\}_{j\geqslant 1}$ in such a way that

$$\begin{aligned} \big |\psi (x_j+\delta _j x_j)-\psi (x_j) - \delta _j x_j \big | \gg M^{\frac{1}{2}}\cdot \big (\delta _j x_j \log (\delta _j^{-1}+2)\big )^{\frac{1}{2}}. \end{aligned}$$

2 Proof of Theorem 1.2

Throughout this section, we will denote by $\varrho =\beta +i\gamma$ the non-trivial zeros of the Riemann zeta function. We recall the Riemann-von Mangoldt formula

$$\begin{aligned} N(T):=\{ \varrho : 0 \leqslant \Im m(\varrho ) \leqslant T \} = \frac{T}{2\pi } \log \frac{T}{2\pi e} +O(\log (T+2)), \end{aligned}$$

(2.1)

which is valid for $T \geqslant 0$.

A major ingredient in our proof is the following explicit formula for $\psi _\eta (x,\delta )$ and a related quantity.

Lemma 2.1

Let $0<\kappa < \frac{1}{2}$ and³$\eta \in {\mathcal {E}}_\kappa$. For $t\geqslant 0$ and $0<\delta < \kappa$ we have the formulas

$$\begin{aligned} \psi _\eta (\mathrm{e}^t,\delta ) - \mathrm{e}^{\frac{t}{2} } \delta {\mathcal {L}}_\eta (\tfrac{\delta }{2})=- \delta \sum _{\varrho } \mathrm{e}^{ (\varrho -\frac{1}{2}) t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big )+O_{\kappa ,\eta }( E_{\kappa ,\eta }(t,\delta )); \end{aligned}$$

(2.2)

$$\begin{aligned} \begin{aligned} \mathrm{e}^{-\frac{t}{2}}\Big ( \sum _{n\geqslant 1} \Lambda (n) \eta ( \delta ^{-1} ( \log n - t ) ) - \mathrm{e}^{t } \delta {\mathcal {L}}_\eta (\delta )\Big )&=- \delta \sum _{\varrho } \mathrm{e}^{ (\varrho -\frac{1}{2}) t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho }{i}\Big )\\&\quad +O_{\kappa ,\eta }\big ( \mathrm{e}^{-\frac{t}{2}}(\delta + E_{\kappa ,\eta }(t,\delta ))\big ), \end{aligned} \end{aligned}$$

(2.3)

where $\varrho$ runs over the nontrivial zeros of $\zeta (s)$, and

$$\begin{aligned} E_{\kappa ,\eta }(t,\delta ):= {\left\{ \begin{array}{ll} \delta \mathrm{e}^{- \frac{ t}{2} }+\log (\delta ^{-1}+2)\mathrm{e}^{-\frac{\kappa t}{\delta } } &{} \text { if } t\geqslant 1 ,\\ \frac{\delta }{t}+\log (\delta ^{-1}+2)\mathrm{e}^{-\frac{\kappa t}{\delta } } &{} \text { if } \delta \leqslant t <1 ,\\ \log (\delta ^{-1}+2) &{} \text { if } 0\leqslant t \leqslant \delta . \end{array}\right. } \end{aligned}$$

(2.4)

Under RH we have the uniform bound

$$\begin{aligned} \psi _\eta (\mathrm{e}^t,\delta ) - \mathrm{e}^{ \frac{t}{2} } \delta {\mathcal {L}}_\eta ( \tfrac{\delta }{2}) \ll _{\kappa ,\eta } \log (\delta ^{-1}+2). \end{aligned}$$

(2.5)

If in addition to RH we assume that ${{\widehat{\eta }}}(s) \ll (1+|s|)^{-2-\varepsilon }$ for some $\varepsilon \geqslant 0$ and whenever $|\Im m(s)|\leqslant \frac{1}{2}$, then we have the estimate

$$\begin{aligned} \begin{aligned} \mathrm{e}^{-\frac{t}{2}}\Big ( \sum _{n\geqslant 1} \Lambda (n) \eta ( \delta ^{-1} ( \log n - t ) )&- \mathrm{e}^{t } \delta {\mathcal {L}}_\eta (\delta )\Big ) = \psi _\eta (\mathrm{e}^t,\delta ) - \mathrm{e}^{\frac{t}{2} } \delta {\mathcal {L}}_\eta (\tfrac{\delta }{2}) \\&+ O_{\kappa ,\eta }\big (\delta ^{\frac{1}{2}+ \frac{\varepsilon }{2(2+\varepsilon )} } \log (\delta ^{-1}+2)+ E_{\kappa ,\eta }(t,\delta )\big ). \end{aligned} \end{aligned}$$

(2.6)

Proof

To show (2.2) we apply [25, Theorem 12.13] with $F(u):=\eta ( \frac{t+2\pi u}{\delta })$, so that ${{\widehat{F}}}(\xi ) = \mathrm{e}^{i \xi t}\frac{\delta }{2\pi }{{\widehat{\eta }}}(\frac{\delta \xi }{2\pi })$. We obtain that

$$\begin{aligned} \psi _{\eta } ( \mathrm{e}^t,\delta )&- \mathrm{e}^{\frac{t}{2} } \delta {\mathcal {L}}_\eta (\tfrac{\delta }{2})+ \delta \sum _{\varrho } \mathrm{e}^{ (\varrho -\frac{1}{2})t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ) \\=&\mathrm{e}^{- \frac{t}{2} }{\delta } \int _{{\mathbb {R}}} \mathrm{e}^{\frac{\delta x}{2} }\eta (x )\mathrm{d}x- \sum _{n\geqslant 1} \frac{\Lambda (n)}{n^{\frac{1}{2}}} \eta \Big ( \frac{t+\log n}{\delta }\Big ) +\Big ( \frac{\Gamma '}{\Gamma }\Big (\frac{1}{4}\Big )-\log \pi \Big )\eta \Big ( \frac{t }{\delta }\Big ) \\&+\int _0^\infty \frac{\mathrm{e}^{-\frac{x}{2}}}{1-\mathrm{e}^{-2x}}\Big \{ 2\eta \Big ( \frac{t }{\delta }\Big )-\eta \Big ( \frac{t+ x}{\delta }\Big )-\eta \Big ( \frac{t- x}{\delta }\Big )\Big \}\mathrm{d}x. \end{aligned}$$

A careful analysis of the second integral yields the bound (2.4) whenever $\eta \in {\mathcal {E}}_\kappa$.

The proof of (2.3) is similar, with the choice $F(u):=\mathrm{e}^{ - \pi u }\eta ( \frac{t+2\pi u}{\delta })$, so that

$$\begin{aligned} \int _{{\mathbb {R}}}F(u) \mathrm{e}^{-(\xi -\frac{1}{2})2\pi u} \mathrm{d}u = \mathrm{e}^{ \xi t}\frac{\delta }{2\pi }{{\widehat{\eta }}}\Big (\frac{\delta \xi }{2\pi i }\Big ). \end{aligned}$$

The uniform bound (2.5) follows from the triangle inequality and a straightforward application of the Riemann-von Mangoldt formula (2.1).

We now move to (2.6). It is sufficient to establish the bound

$$\begin{aligned} \delta \sum _{\varrho } \mathrm{e}^{ \varrho t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho }{i}\Big )-\delta \sum _{\varrho } \mathrm{e}^{ \varrho t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ) \ll _{\kappa ,\eta } \mathrm{e}^{\frac{t}{2}}\delta ^{\frac{1}{2}+ \frac{\varepsilon }{2(2+\varepsilon )} } \log (\delta ^{-1}+2). \end{aligned}$$

To show this, we first truncate the infinite sums. Our conditions on $\eta$ imply that

$$\begin{aligned} \delta \sum _{|\varrho |> \delta ^{- \frac{3+\varepsilon }{2+\varepsilon }}} \mathrm{e}^{ \varrho t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho }{i}\Big )-\delta \sum _{|\varrho |> \delta ^{-\frac{3+\varepsilon }{2+\varepsilon }}} \mathrm{e}^{ \varrho t} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ) \ll _{\kappa ,\eta } \mathrm{e}^{\frac{t}{2}}\delta ^{\frac{1}{2}+ \frac{\varepsilon }{2(2+\varepsilon )} } \log (\delta ^{-1}+2). \end{aligned}$$

The rest of the sums over $\varrho$ is bounded by combining (2.1) with the bound

$$\begin{aligned} \begin{aligned} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ) - {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi } \frac{\varrho }{i}\Big )&=\int _{{\mathbb {R}}} (\mathrm{e}^{ \delta (\varrho -\frac{1}{2}) \xi } - \mathrm{e}^{ \delta \varrho \xi }) \eta (\xi ) \mathrm{d}\xi \\&\ll \int _{|\xi |\leqslant \delta ^{-1} } \delta |\xi \eta (\xi )| d\xi + \int _{|\xi | > \delta ^{-1} } \mathrm{e}^{\frac{\delta |\xi |}{2}} | \eta (\xi )| d\xi \\&\ll _{\kappa ,\eta } \delta +\frac{\mathrm{e}^{-\delta ^{-1} \kappa }}{ \kappa -\frac{\delta }{2}} \ll _\kappa \delta . \end{aligned}\end{aligned}$$

(2.7)

$\square$

The following estimate on a convergent sum over zeros will be helpful in calculating the main terms in our lower bounds on moments.

Lemma 2.2

Let $0<\kappa <\frac{1}{2}$, and let $h:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a measurable function such that for all $\xi \in {\mathbb {R}}$, $0\leqslant h(\xi )\ll (|\xi |+1)^{-2}(\log (|\xi |+2))^{-2-\kappa }$, and⁴ for all $t\in {\mathbb {R}}$, ${{\widehat{h}}}(t),{{\widehat{h}}}'(t) \ll \mathrm{e}^{-\kappa |t|}$. For $0<\delta <2\kappa$ we have that

$$\begin{aligned} \sum _{\begin{array}{c} \varrho \end{array}} h\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big )= \alpha (h)\delta ^{-1} \log (\delta ^{-1})+\beta (h) \delta ^{-1}+O_{\kappa ,h}(1), \end{aligned}$$

(2.8)

where $\varrho$ is running over the non-trivial zeros of the Riemann zeta function, and where h is extended to $\{s\in {\mathbb {C}}: |\Im m(s)| < \frac{\kappa }{2\pi } \}$ by writing

$$\begin{aligned} h(z):= \int _{{\mathbb {R}}} \mathrm{e}^{2\pi i z \xi } {{\widehat{h}}}(\xi ) \mathrm{d}\xi . \end{aligned}$$

(2.9)

Proof

The claimed estimate can be established with a slightly weaker error term (and a different class of functions h) using the Riemann-von Mangoldt formula (2.1) and the bound

$$\begin{aligned} h\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ) - h\Big (\frac{\delta \Im m(\varrho ) }{2\pi }\Big )\ll _{\kappa ,h} \delta , \end{aligned}$$

(2.10)

which follows from a calculation similar to (2.7). To obtain the claimed error term, we will use a different technique. Applying the explicit formula [25, Theorem 12.13] with $F(x):=2\pi \delta ^{-1} {{\widehat{h}}}(-2\pi \delta ^{-1} x),$ we obtain that

$$\begin{aligned} \sum _{\begin{array}{c} \varrho \end{array}} h\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big )=\delta ^{-1}\big (b_1(h)+b_2(h)+I(h)\big )+h\Big (\frac{i\delta }{4\pi }\Big )+h\Big (-\frac{i\delta }{4\pi }\Big ),\nonumber \\\end{aligned}$$

(2.11)

where

$$\begin{aligned} b_1(h):= & {} \Big (\frac{\Gamma '}{\Gamma }\Big (\frac{1}{4}\Big )-\log \pi \Big ){{\widehat{h}}}(0);\\ b_2(h):= & {} -\sum _{n=1}^\infty \frac{\Lambda (n)}{ n^{\frac{1}{2}}}\big ({{\widehat{h}}}(\delta ^{-1}\log n)+\widehat{h}(-\delta ^{-1}\log n)\big ); \\ I(h):= & {} \int _0^\infty \frac{\mathrm{e}^{-\frac{x}{2} }}{1-\mathrm{e}^{-2 x}}\big (2{{\widehat{h}}}(0)-{{\widehat{h}}}(\delta ^{-1} x)-{{\widehat{h}}}(-\delta ^{-1} x)\big )\mathrm{d}x. \end{aligned}$$

Integration by parts shows that $b_2(h) \ll _\kappa 2^{-\kappa \delta ^{-1}}$. We split the integral I(h) into the three ranges $[0,\delta ]$, $[ \delta ,1]$, $[1,+\infty )$, and denote by $I_1(h),I_2(h),I_3(h)$ the respective integrals. We have that

$$\begin{aligned} I_3(h) = {{\widehat{h}}}(0)\int _1^\infty \frac{2\mathrm{e}^{-\frac{x}{2} }}{1-\mathrm{e}^{-2 x}}\mathrm{d}x+O_h(\mathrm{e}^{-\frac{\kappa }{\delta }}). \end{aligned}$$

Moreover,

$$\begin{aligned} I_2(h)&={{\widehat{h}}}(0)\log (\delta ^{-1})+ {{\widehat{h}}}(0)\int _0 ^1\!\!\Big ( \frac{2\mathrm{e}^{-\frac{x}{2} }}{1-\mathrm{e}^{-2 x}}-\frac{1}{x}\Big )\mathrm{d}x\\&\quad -\int _{{\mathbb {R}}} h(\xi )\int _1^\infty \!\!\!\cos (2\pi x \xi )\frac{\mathrm{d}x}{x}\mathrm{d}\xi +O_h(\delta ). \end{aligned}$$

As for $I_1(h)$, we obtain that

$$\begin{aligned} I_1(h)= \int _{{\mathbb {R}}} h(\xi )\int _0^1 (1-\cos ({2\pi x\xi } )) \frac{ \mathrm{d}x}{x} \mathrm{d}\xi +O_h(\delta ). \end{aligned}$$

Collecting our estimates for $I_1(h),I_2(h),I_3(h)$ as well as the estimate $h(\pm \frac{i\delta }{4\pi }) = h(0)+O_h(\delta )$, we deduce that

$$\begin{aligned} \delta \sum _{\begin{array}{c} \varrho \end{array}} h\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big )= {{\widehat{h}}}(0)\big (\log (\delta ^{-1})+ C\big ) +\int _{{\mathbb {R}}} h(\xi ) \log |\xi | \mathrm{d}\xi +O_{\kappa ,h}(\delta ), \end{aligned}$$

where

$$\begin{aligned} C&:= \int _0 ^1\Big ( \frac{2\mathrm{e}^{-\frac{x}{2} }}{1-\mathrm{e}^{-2 x}}-\frac{1}{x}\Big )\mathrm{d}x +\int _1^\infty \frac{2\mathrm{e}^{-\frac{x}{2} }}{1-\mathrm{e}^{-2 x}}\mathrm{d}x + \frac{\Gamma '}{\Gamma }\Big (\frac{1}{4}\Big )-\log \pi \\&\quad + \int _0^1 (1-\cos (2\pi x ))\frac{\mathrm{d}x}{x}-\int _1^\infty \cos (2\pi x )\frac{\mathrm{d}x}{x} . \end{aligned}$$

We will show that $C=0$, from which the claimed estimate follows. We have the identity [35, §II.0, Exercise 149]

$$\begin{aligned} \frac{\Gamma '}{\Gamma }\Big (\frac{1}{4}\Big )= \int _0^\infty \Big (\frac{ \mathrm{e}^{-2x }}{x}- \frac{2\mathrm{e}^{-\frac{x}{2} }}{1-\mathrm{e}^{-2 x}}\Big ) \mathrm{d}x. \end{aligned}$$

We deduce that

$$\begin{aligned} C= \int _0^\infty \frac{ \mathrm{e}^{-2x} - \cos (2x)}{x}\mathrm{d}x, \end{aligned}$$

which is readily shown to be equal to zero using the residue theorem. $\square$

We will also need the following combinatorial lemma.

Lemma 2.3

Let $0<\kappa < \frac{1}{2}$, $\eta \in {\mathcal {E}}_\kappa$, and assume⁵ RH. For $\delta \in (0,\kappa )$, $m\in {\mathbb {N}}$, and in the range $m\leqslant \delta ^{-\frac{1}{2}}(\log (\delta ^{-1}+2))^{\frac{1}{2}}$, we have the lower bound

$$\begin{aligned}&\delta ^{2m}\!\!\!\!\!\!\!\sum _{\begin{array}{c} \gamma _1,\dots ,\gamma _{2m} \\ \gamma _1+\dots +\gamma _{2m}=0 \end{array}} {{\widehat{\eta }}}\Big (\frac{\delta \gamma _1}{2\pi }\Big )\cdots {{\widehat{\eta }}}\Big (\frac{\delta \gamma _{2m}}{2\pi }\Big )\\&\quad \geqslant \mu _{2m} \delta ^m \big (\alpha ({{\widehat{\eta }}}^2) \log (\delta ^{-1})+\beta ({{\widehat{\eta }}}^2)\big )^{m}\Big (1+O_{\kappa ,\eta }\Big (\frac{ m^2 \delta }{ \log (\delta ^{-1}+2)}\Big )\!\Big ), \end{aligned}$$

where the $\gamma _j$ are running over the imaginary parts of the non-trivial zeros of the Riemann zeta function.

Proof

We will show that

$$\begin{aligned} M_{2m}:=\sum _{\begin{array}{c} \gamma _1,\dots ,\gamma _{2m} \\ \gamma _1+\dots +\gamma _{2m}=0 \end{array}} {{\widehat{\eta }}}\Big (\frac{\delta \gamma _1}{2\pi }\Big )\cdots {{\widehat{\eta }}}\Big (\frac{\delta \gamma _{2m}}{2\pi }\Big )\geqslant \mu _{2m}\big ( s_2^{m}-m(m-1)s_2^{m-2}s_4\big ),\nonumber \\ \end{aligned}$$

(2.12)

where $s_{2j}:= \sum _{\begin{array}{c} \gamma \end{array}} |{{\widehat{\eta }}}(\frac{\delta \gamma }{2\pi })|^{2j}$. Combining this bound with Lemma 2.2 with $h=|{{\widehat{\eta }}}|^2={{\widehat{\eta }}}^2$ and $h=|{{\widehat{\eta }}}|^4={{\widehat{\eta }}}^4$ implies the claimed bound. One can check that $\eta \in {\mathcal {E}}_\kappa$ implies that for both those choices of h, we have the bounds ${{\widehat{h}}}(t), {{\widehat{h}}}'(t) \ll (|t|^3+1) \mathrm{e}^{-\kappa |t|}$.

Now, to establish (2.12), note that this is an equality for $m=1$, and is clear for $m=2$. In the general case, we have that

$$\begin{aligned} M_{2m}\geqslant M_{2m}':=\sum _{\begin{array}{c} \gamma _1,\dots ,\gamma _{2m} \text { distinct} \\ \gamma _1+\dots +\gamma _{2m}=0 \end{array}} {{\widehat{\eta }}}\Big (\frac{\delta \gamma _1}{2\pi }\Big )\cdots {{\widehat{\eta }}}\Big (\frac{\delta \gamma _{2m}}{2\pi }\Big ). \end{aligned}$$

Note that $M_2=M_2'=s_2$. One can restrict the sum in $M'_{2m}$ to those 2m-tuples of zeros for which for each $1\leqslant j\leqslant 2m$, there exists $1\leqslant i \leqslant 2m$, $i\ne j$, such that $\gamma _i=-\gamma _j$. In other words, for each involution $\pi : \{ 1, \dots ,2m\} \rightarrow \{ 1, \dots ,2m\}$ with no fixed points, there exists a subset of 2m-tuples of zeros $\gamma _1,\dots \gamma _{2m}$ such that for each $1\leqslant j \leqslant 2m$, $\gamma _{j} = -\gamma _{\pi (j)}$. Note also that since the $\gamma _j$ are distinct in $M'_{2m}$, the sets of 2m-tuples associated to different involutions $\pi$ are distinct. Since the total number of such involutions is equal to $\mu _{2m}$, it follows that

$$\begin{aligned} M_{2m}'&= \mu _{2m} \sum \limits _{\gamma _1}^{}\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _1 }{2\pi }\Big ) \Big |^2 \sum \limits _{\gamma _3\notin \{ \gamma _1,-\gamma _1\}}^{}\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _3 }{2\pi }\Big ) \Big |^2\ldots \\&\sum \limits _{\gamma _{2m-1}\notin \{ \gamma _1,-\gamma _1,\ldots , \gamma _{2m-3},-\gamma _{2m-3}\}}^{}\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _{2m-1} }{2\pi }\Big ) \Big |^2. \end{aligned}$$

Therefore, by symmetry we have that

$$\begin{aligned}\frac{M_{2m}'}{\mu _{2m}}&=\sum _{\begin{array}{c} \gamma _1 \end{array}}\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _1 }{2\pi }\Big ) \Big |^2 \sum _{\begin{array}{c} \gamma _3\notin \{ \gamma _1,-\gamma _1\} \end{array}}\!\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _3 }{2\pi }\Big ) \Big |^2\ldots \Bigg \{ s_2-2\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _1 }{2\pi }\Big ) \Big |^2\!\!-\ldots -2\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _{2m-3} }{2\pi }\Big ) \Big |^2\Bigg \} \\&= \sum _{\begin{array}{c} \gamma _1 \end{array}}\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _1 }{2\pi }\Big ) \Big |^2 \sum _{\begin{array}{c} \gamma _3\notin \{ \gamma _1,-\gamma _1\} \end{array}}\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _3 }{2\pi }\Big ) \Big |^2\ldots \Bigg \{ s_2 -2(m-1)\Big |{{\widehat{\eta }}}\Big (\frac{\delta \gamma _{2m-3} }{2\pi }\Big ) \Big |^2\Bigg \} \\&\geqslant \frac{M_{2m-2}'}{\mu _{2(m-1)}}s_2 -2(m-1)s_2^{m-2}s_4. \end{aligned}$$

The claimed bound follows by induction on m. $\square$

We are ready to prove our main theorem.

Proof of Theorem 1.2

We begin by applying Lemma 2.1. Under RH, we set $T:=\log X$ and obtain that

$$\begin{aligned} (-1)^n M_n(\mathrm{e}^T,\delta ;\eta ,\Phi )&=\frac{(-1)^n}{ T\int _0^\infty \Phi }\int _{0}^{\infty } \Phi \Big ( \frac{t}{T}\Big )\big (\psi _\eta (\mathrm{e}^t,\delta )-\mathrm{e}^{\frac{t}{2}}\delta {\mathcal {L}}_\eta ( \tfrac{\delta }{2} )\big )^{n} \mathrm{d}t \\&= \frac{\delta ^n}{\int _0^\infty \Phi } \sum _{\gamma _1,\dots ,\gamma _n} {{\widehat{\eta }}}\Big (\frac{\delta \gamma _1}{2\pi }\Big )\cdots {{\widehat{\eta }}}\Big (\frac{\delta \gamma _n}{2\pi }\Big )\int _0^{\infty } \mathrm{e}^{i tT (\gamma _1+\dots +\gamma _n) }\Phi (t) \mathrm{d}t\\&\quad +O\Big (\frac{ \delta (K_{\eta } \log (\delta ^{-1}+2))^n}{T} \Big )\\&= \frac{ \delta ^n}{2\int _0^\infty \Phi } \sum _{\gamma _1,\dots ,\gamma _n} {{\widehat{\Phi }}}\Big (\frac{T(\gamma _1+\dots +\gamma _n)}{2\pi }\Big ){{\widehat{\eta }}}\Big (\frac{\delta \gamma _1}{2\pi }\Big )\cdots {{\widehat{\eta }}}\Big (\frac{\delta \gamma _n}{2\pi }\Big )\\&\quad +O\Big (\frac{ \delta (K_{\eta } \log (\delta ^{-1}+2))^n}{T} \Big ), \end{aligned}$$

since both $\Phi$ and ${{\widehat{\Phi }}}$ are even and real-valued. Here, $\gamma _1,\dots , \gamma _n$ are running over the imaginary parts of the non-trivial zeros of $\zeta (s)$. If n is odd, then the claimed estimate follows from discarding the sum over zeros entirely. If n is even, then by positivity of ${{\widehat{\eta }}}$ and ${{\widehat{\Phi }}}$ we may only keep the terms for which $\gamma _1+\dots +\gamma _n=0$, and apply Lemma 2.3. The claimed lower bound follows. $\square$

3 Proof of Corollaries 1.3 and 1.4

We first need to establish the following proposition, which is strongly inspired from the work of Kaczorowski and Pintz [18]. We consider

$$\begin{aligned} F(x,\delta ;\eta ):=- \delta \sum _{\varrho } \frac{x^{ \varrho -\frac{1}{2} }}{\varrho -\frac{1}{2}} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ), \end{aligned}$$

which is readily shown to be real-valued by grouping conjugate zeros.

Proposition 3.1

Assume that RH is false, and let $\eta \in {\mathcal {E}}_\kappa$ with $0<\kappa < \frac{1}{2}$. Then, there exists an absolute (ineffective) constant $\theta >0$ and a sequence $\{x_j\}_{j\geqslant 1}$ tending to infinity such that for each $j\geqslant 1$ and uniformly for $x_j^{-\theta }\leqslant \delta \leqslant \delta _{\eta }$, where $\delta _{\eta }>0$ is small enough, we have that

$$\begin{aligned} F(x_j,\delta ;\eta ) > x_j^{\theta }. \end{aligned}$$

Proof

Consider, for $\Theta >0$, the $(n-1)$-fold average

$$\begin{aligned} F_n(\mathrm{e}^t,\delta ,\Theta ;\eta ):= - \delta \sum _{\varrho } \frac{\mathrm{e}^{ ( \varrho -\frac{1}{2})t }}{(\varrho -\frac{1}{2})^n} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big ) -\delta \frac{\mathrm{e}^{\Theta t}}{\Theta ^{n-1}}, \end{aligned}$$

so that $\frac{\mathrm{d}^{n-1}}{(\mathrm{d}t)^{n-1}}F_n(\mathrm{e}^t,\delta ,\Theta ;\eta )= F(\mathrm{e}^t,\delta ;\eta )-\delta \mathrm{e}^{\Theta t}$. Let $\varrho _e=\beta _e+i\gamma _e$ be a zero of $\zeta (s)$ violating RH, of least positive imaginary part $\gamma _e$, and such that there is no other zero of imaginary part equal to $\gamma _e$ but of greater real part. Let moreover $\varepsilon < \beta _e-\frac{1}{2}$. We will show that $F_n(\mathrm{e}^t,\delta ,\Theta ;\eta )=0$ for many values of t (independently of $\delta$), and then apply Rolle’s theorem.

We pick $t=cn$, with $n\geqslant 1$ and $c\in {\mathbb {R}}$. If $\Theta \leqslant \varepsilon$ and c is large enough in terms of $\varepsilon$ and $\Theta$, say $c\geqslant c_0(\varepsilon )$ (later we will require that $c_0(\varepsilon )\geqslant 1$), then

$$\begin{aligned} \frac{\mathrm{e}^{cn\Theta }}{\Theta ^{n-1}} < \Bigg (\frac{\mathrm{e}^{c(\beta _e-\frac{1}{2})}}{2|\varrho _e-\frac{1}{2}|}\Bigg )^n. \end{aligned}$$

We will also impose c to be bounded in terms of $\varepsilon$ and $\varrho _e$, say $c\leqslant c_1(\varepsilon )$. More precisely, we pick $c_1(\varepsilon )= c_0(\varepsilon ) + 2$. Then, there exists $U_\varepsilon$ large enough so that

$$\begin{aligned} \sum _{|\Im m(\varrho )|>U_\varepsilon } \frac{\mathrm{e}^{ c n (\varrho -\frac{1}{2}) }}{(\varrho -\frac{1}{2})^n} {{\widehat{\eta }}}\Bigg (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Bigg ) \ll _{\kappa ,\eta } (\log U_\varepsilon ) \frac{\mathrm{e}^{c\frac{n}{2}}}{ U_\varepsilon ^{n-1} }< \Bigg (\frac{\mathrm{e}^{c(\beta _e-\frac{1}{2})}}{2|\varrho _e-\frac{1}{2}|}\Bigg )^n, \end{aligned}$$

whenever $\delta \leqslant \kappa$, $n>n_0(\varepsilon )$ and $c_0(\varepsilon )< c < c_1(\varepsilon )$. Here we used the bound

$$\begin{aligned} {{\widehat{\eta }}}(s)&= \int _{{\mathbb {R}}} \mathrm{e}^{ -2\pi i s x} \eta (x)\mathrm{d}x \\&\quad \ll \int _{0}^{\infty } \mathrm{e}^{ 2\pi |\Im m(s)| x} \mathrm{e}^{-\kappa x}\mathrm{d}x\ll \frac{1}{ \kappa - 2\pi |\Im m(s)|}\qquad (|\Im m(s)| < \kappa / 2\pi ) . \end{aligned}$$

We conclude that under these last two conditions,

$$\begin{aligned} F_n(\mathrm{e}^{cn},\delta ,\Theta ;\eta ) = -\delta \sum _{|\Im m(\varrho )|\leqslant U_\varepsilon } \frac{\mathrm{e}^{ c n (\varrho -\frac{1}{2}) }}{(\varrho -\frac{1}{2})^n} {{\widehat{\eta }}}\Bigg (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Bigg ) +O\Bigg (\delta \Bigg (\frac{\mathrm{e}^{c(\beta _e-\frac{1}{2})}}{2|\varrho _e-\frac{1}{2}|}\Bigg )^n\Bigg ). \end{aligned}$$

For two distinct zeros $\varrho _1,\varrho _2$ of $\zeta (s)$ of positive imaginary part at most $U_\varepsilon$, consider the function

$$\begin{aligned} f: (c_0(\varepsilon ),c_1(\varepsilon ))&\rightarrow {\mathbb {R}}\\ c&\mapsto c(\Re e(\varrho _1)\!-\!\tfrac{1}{2})-\log |\varrho _1-\tfrac{1}{2}|\!-\!c(\Re e(\varrho _2)-\tfrac{1}{2})+\log |\varrho _2-\tfrac{1}{2}|. \end{aligned}$$

This linear function is not identically zero and has at most one zero, hence there exists a subset $S_1 \subset (c_0(\varepsilon ),c_1(\varepsilon ))$ which is a union of two intervals such that for all $c\in S_1$, $|f(c)|\geqslant \kappa _\varepsilon$, for some fixed and small enough $\kappa _\varepsilon >0$. By picking $\kappa _\varepsilon$ small enough, we may require that $\lambda (S_1) \geqslant 2-2^{-\#\{ \varrho \, :\, \zeta (\varrho )=0,\, |\Im m(\varrho )| \leqslant U_\varepsilon \}}$, where $\lambda$ is the Lebesgue measure. We may iterate this procedure with all pairs of distinct zeros $\varrho _j,\varrho _k$ such that $0<\Im m(\varrho _j),\Im m(\varrho _k) \leqslant U_\varepsilon$, and deduce that there exists a subset $S\subset (c_0(\varepsilon ),c_1(\varepsilon ))$ of measure $\geqslant 1$ which is a disjoint union of at most $2^{\#\{ \varrho \, :\, \zeta (\varrho )=0,\, 0 < \Im m(\varrho ) \leqslant U_\varepsilon \}} +1$ intervals $(\alpha _j,\tau _j)$ such that for each j and whenever $c\in (\alpha _j,\tau _j)$, there exists a zero $\varrho _j=\beta _j+i\gamma _j$ such that

$$\begin{aligned}&c(\Re e(\varrho _j)-\tfrac{1}{2})-\log |\varrho _j-\tfrac{1}{2}| - \max \{ c(\Re e(\varrho )-\tfrac{1}{2})-\log |\varrho -\tfrac{1}{2}| : \zeta (\varrho )\\&\quad =0,0<\Im m(\varrho )\leqslant U_\varepsilon \}\geqslant \kappa _\varepsilon . \end{aligned}$$

Then, denoting by $m_j$ the multiplicity of $\varrho _j$, for all $c\in (\alpha _j,\tau _j)$ we have that

$$\begin{aligned} F_n(\mathrm{e}^{cn},\delta ,\Theta ;\eta )&= -\delta m_j \Re e\Bigg (\frac{\mathrm{e}^{ c n (\varrho _j-\frac{1}{2}) }}{(\varrho _j-\frac{1}{2})^n} {{\widehat{\eta }}}\Bigg (\frac{\delta }{2\pi }\frac{\varrho _j-\frac{1}{2}}{i}\Bigg ) \Bigg )\\&\quad +O\Bigg (\delta \Bigg (\frac{K_\varepsilon \mathrm{e}^{c(\beta _j-\frac{1}{2})}}{|\varrho _j-\frac{1}{2}|}\Bigg )^n\Bigg ), \end{aligned}$$

where $0<K_\varepsilon <1$ is absolute. Note that for all small enough $\delta$ and for all j, we have that ${{\widehat{\eta }}}(\frac{\delta }{2\pi }\frac{\varrho _j-\frac{1}{2}}{i}) = {{\widehat{\eta }}}(0)+O(\delta ).$ Hence,

$$\begin{aligned} F_n(\mathrm{e}^{cn},\delta ,\Theta ;\eta )&= -\delta m_j \Re e\Bigg (\frac{\mathrm{e}^{ cn (\varrho _j-\frac{1}{2}) }}{(\varrho _j-\frac{1}{2})^n} \Bigg ){{\widehat{\eta }}}(0)\\&\quad +O\Bigg (\delta ^2 m_j\Bigg (\frac{\mathrm{e}^{c(\beta _j-\frac{1}{2})}}{|\varrho _j-\frac{1}{2}|}\Bigg )^n+\delta \Bigg (\frac{K_\varepsilon \mathrm{e}^{c(\beta _j-\frac{1}{2})}}{|\varrho _j-\frac{1}{2}|}\Bigg )^n\Bigg ). \end{aligned}$$

For n large enough, this function has at least $(\tau _j-\alpha _j)\Im m(\varrho _j) n/\pi +O(1) \geqslant 4 (\tau _j-\alpha _j)n$ zeros for $c\in (\alpha _j,\tau _j)$. Indeed, this follows from the intermediate value theorem combined with the identity

$$\begin{aligned} \Re e\Bigg (\frac{\mathrm{e}^{ cn (\varrho _j-\frac{1}{2}) }}{(\varrho _j-\frac{1}{2})^n} \Bigg ) = \frac{\mathrm{e}^{ cn(\beta _j -\frac{1}{2})}}{|\varrho _j-\frac{1}{2}|^n} \cos (\nu _{j,c} n), \end{aligned}$$

where $\nu _{j,c}:=\Im m(\varrho _j) c-\Im m (\log (\varrho _j-\frac{1}{2}))$. Since this is true for every j, we conclude that $F_n(\mathrm{e}^{cn},\delta ,\Theta ;\eta )$ has at least $4 n \lambda (S) \geqslant 4n$ zeros for $c\in S$. In other words, $F_n(\mathrm{e}^t,\delta ,\Theta ;\eta )$ has at least 4n zeros for $t\in [c_0(\varepsilon ) n , c_1(\varepsilon ) n ]$. By Rolle’s theorem, we deduce that $F(\mathrm{e}^t,\delta ;\eta ) -\delta \mathrm{e}^{\Theta t}$ has at least 3n zeros on this interval (note that by our conditions on $\eta$, $F(\mathrm{e}^t,\delta ;\eta )$ is continuous). In the range $\mathrm{e}^{-\theta t}\leqslant \delta$, the result follows whenever $0<\theta <\Theta /2$. $\square$

We are ready to prove our first unconditional result.

Proof of Corollary 1.3

If RH is true, then this is a particular case of Theorem 1.2. Let us then assume that RH is false. By Hölder’s inequality we have that

$$\begin{aligned} M_{2m}(X,\delta ;\eta ,\Phi ) ^{\frac{1}{2m}}&\geqslant \frac{1}{(\log X)\int _0^\infty \Phi } \int _1^{\infty } \Phi \Big ( \frac{\log x}{\log X} \Big )\big |\psi _{\eta }(x,\delta )-x^{\frac{1}{2} } \delta {\mathcal {L}}_\eta (\tfrac{\delta }{2})\big | \frac{\mathrm{d}x}{x} \\&\geqslant \frac{c(\Phi )}{(\log X)\int _0^\infty \Phi } \int _1^{ X^{\kappa (\Phi )}} \big (\psi _{\eta }(x,\delta )-x^{\frac{1}{2} } \delta {\mathcal {L}}_\eta (\tfrac{\delta }{2})\big ) \frac{\mathrm{d}x}{x}, \end{aligned}$$

where $c(\Phi ),\kappa (\Phi )>0$. By Lemma 2.1, the integral is equal to

$$\begin{aligned} - \delta \sum _{\varrho } \frac{X^{\kappa (\Phi ) (\varrho -\frac{1}{2}) }}{\varrho -\frac{1}{2}} {{\widehat{\eta }}}\Big (\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i}\Big )+O_{\Phi ,\eta }\big ( {\delta }(\log (\delta ^{-1}+2))^2\big ), \end{aligned}$$

by the Riemann-von Mangoldt formula (2.1). The claimed $\Omega$-result then follows from Proposition 3.1. $\square$

In order to prove Corollary 1.4, we will apply Theorem 1.2 with $\eta (u)= \max (0,1-|u|)$. This is not an element of ${\mathcal {E}}_\kappa$ since it is not differentiable. However, as mentioned in its statement, one can go through the proof of Lemma 2.1 and check that it applies when $\eta$ is Lipschitz, compactly supported, and monotonic on ${\mathbb {R}}_{\geqslant 0}$; we deduce that the same is true for Theorem 1.2 (note that the conditions of Lemma 2.2 are satisfied for $h={{\widehat{\eta }}}^2$).

Proof of Corollary 1.4

If RH is false, then the result follows from an adaptation of the proof of Proposition 3.1. Rather than going through the proof, we highlight the two major differences. Firstly, the function we need to study is

$$\begin{aligned} - \sum _{\varrho } \frac{\mathrm{e}^{ \varrho t }}{\varrho ^n} ((1+\delta )^\varrho -1) -\delta \frac{\mathrm{e}^{(\frac{1}{2}+\Theta ) t}}{(\frac{1}{2}+\Theta )^{n-1}}, \end{aligned}$$

which has the weight $(1+\delta )^\varrho -1$ instead of $\delta {{\widehat{\eta }}}(\frac{\delta }{2\pi }\frac{\varrho -\frac{1}{2}}{i})$. However, this weight is $\ll \delta |\varrho |$ uniformly for all $0<\delta \leqslant 1$ and $0<\Re e(\varrho )<1$. The second major difference is the proof that the existence of two zeros of the continuous and piecewise differentiable function

$$\begin{aligned} - \sum _{\varrho } \frac{\mathrm{e}^{ \varrho t }}{\varrho ^2} \big ((1+\delta )^\varrho -1\big ) -\delta \frac{\mathrm{e}^{(\frac{1}{2}+\Theta )t}}{\frac{1}{2}+\Theta } \end{aligned}$$

implies that the piecewise continuous function

$$\begin{aligned} - \sum _{\varrho } \frac{\mathrm{e}^{ \varrho t }}{\varrho } \big ((1+\delta )^\varrho -1\big ) -\delta \mathrm{e}^{(\frac{1}{2}+\Theta ) t} \end{aligned}$$

has at least one non-negative value between those zeros. This can be done using a straightforward generalization of Rolle’s theorem, which states that if f is continuous on [a, b] for which $f(a)=f(b)$ and the one-sided derivatives

$$\begin{aligned} f^{\pm }(c):=\lim _{x\rightarrow c^{\pm }} \frac{f(x)-f(c)}{x-c} \end{aligned}$$

exist for all $c\in (a,b)$, then there exists $c_0\in (a,b)$ such that $f^+(c_0)f^-(c_0)\leqslant 0$. The rest of the proof is similar.

We now assume RH. Let us also assume that for all large enough x and for all $\delta '$ in the range $\frac{\varepsilon _0(\log _3 x)^{\frac{9}{2}}}{4(\log x)^{2} (\log _2 x)^{\frac{5}{2}}} \leqslant \delta ' \leqslant 2\frac{(\log _3 x)^3}{(\log _2 x)^{2}}$ we have that

$$\begin{aligned} \big |\psi (x+\delta ' x)-\psi (x) - \delta ' x \big | \leqslant \varepsilon _0 \delta '^{-\frac{1}{4}}(\log (\delta '^{-1}+2))^{\frac{1}{4}} \cdot \big (\delta ' x \log (\delta '^{-1}+2)\big )^{\frac{1}{2}}, \end{aligned}$$

where $\varepsilon _0>0$ is the implied constant in the first error term in (1.6).

Define $\eta (u):= \max (0,1-|u|)$, which is even, non-negative, compactly supported and monotonic for $u\geqslant 0$. Moreover, ${{\widehat{\eta }}} (\xi ) = (\sin (\pi \xi )/(\pi \xi ))^2\geqslant 0$. Now, for any $0<\delta \leqslant 1$, $x\geqslant 1$ and $x\mathrm{e}^{-\delta }\leqslant n\leqslant x\mathrm{e}^\delta$, we write $\eta (\delta ^{-1} \log (\frac{n}{x})) = 1-\delta ^{-1}|\int _n^x \frac{\mathrm{d}t}{t}|$ and deduce that

$$\begin{aligned} \sum _{n\geqslant 1} \Lambda (n)&\eta \Bigg (\delta ^{-1} \log \Big ( \frac{n}{x}\Bigg ) \Bigg ) -x \delta {\mathcal {L}}_\eta (\delta ) = \psi (x\mathrm{e}^\delta )-\psi (x\mathrm{e}^{-\delta })\\&\quad - \delta ^{-1}\Bigg ( \int _x^{x\mathrm{e}^\delta } \Bigg ( \sum _{ t< n\leqslant x\mathrm{e}^\delta } \Lambda (n) \Bigg ) \frac{\mathrm{d}t}{t} \\&\quad + \int _{x\mathrm{e}^{-\delta }}^x \Bigg ( \sum _{ x\mathrm{e}^{-\delta }< n\leqslant t } \Lambda (n) \Bigg ) \frac{\mathrm{d}t}{t} \Bigg )- x\delta \int _{{\mathbb {R}}} \eta (u)\mathrm{e}^{\delta u}\mathrm{d}u \\&= \psi (x\mathrm{e}^\delta )-\psi (x\mathrm{e}^{-\delta })-2x\sinh (\delta ) \\&\quad -\delta ^{-1}\Bigg ( \int _x^{x \mathrm{e}^A} \bigg (\psi (x\mathrm{e}^\delta )-\psi (t)-(x\mathrm{e}^\delta -t )\bigg ) \frac{\mathrm{d}t}{t} \\&\quad + \int _{x\mathrm{e}^{-A}}^x \bigg (\psi (t)-\psi ( x\mathrm{e}^{-\delta })-(t-x\mathrm{e}^{-\delta } )\bigg ) \frac{\mathrm{d}t}{t} \Bigg )\\&\quad +O\bigg (\delta ^{-1} x^{\frac{1}{2}}(\log x)^2 (\delta -A) \bigg ), \end{aligned}$$

for any $0<A<\delta$; in particular for $A=\delta - \varepsilon _0\delta ^{\frac{5}{4}}(\log (\delta ^{-1}+2))^{\frac{3}{4}}/(\log x)^2$. Here we used the (trivial) RH bound

$$\begin{aligned} \psi (M)-\psi (N)-(M-N) \ll M^{\frac{1}{2}} (\log (M+2))^2 \quad ( 1\leqslant N\leqslant M). \end{aligned}$$

By our hypothesis, we deduce that for X large enough, $m\geqslant 2$, $\delta = (\log _3 X )^3/(\log _2 X)^2$ and in the range $\exp ((\log X)^{\frac{1}{2}}) \leqslant x \leqslant X$,

$$\begin{aligned} x^{-\frac{1}{2}} \Big ( \sum _{n\geqslant 1} \Lambda (n) \eta \Big (\delta ^{-1} \log \Big ( \frac{n}{x}\Big ) \Big ) - x \delta {\mathcal {L}}_\eta (\delta )\Big ) \ll \varepsilon _0 \delta ^{\frac{1}{4}} \big (\log (\delta ^{-1}+2)\big )^{\frac{3}{4}}. \end{aligned}$$

Combining this with (2.6) with $\varepsilon =0$ (since ${{\widehat{\eta }}} (s) \ll (1+|s|)^{-2}$ for $|\Im m(s)|\leqslant \frac{1}{2}$, and recalling that the differentiability condition in Lemma 2.1 can be replaced by one of Lipschitz since $\eta$ has compact support and is decreasing on ${\mathbb {R}}_{\geqslant 0}$), we deduce that

$$\begin{aligned} \psi _\eta (x,\delta ) - x^{\frac{1}{2}} \delta {\mathcal {L}}_\eta (\tfrac{\delta }{2} )\ll \varepsilon _0 \delta ^{\frac{1}{4}} \big (\log (\delta ^{-1}+2)\big )^{\frac{3}{4}}. \end{aligned}$$

Now, making the choice $\Phi =\eta$, this implies that for X large enough,

$$\begin{aligned} M_n(X,\delta ;\eta ,\Phi )&=\frac{1}{ (\log X) \int _{0}^{\infty }\Phi }\int _{\exp ((\log X)^{\frac{1}{2}})}^{\infty } \Phi \Big ( \frac{\log x}{\log X}\Big )\big (\psi _\eta (x,\delta ;\eta )\\&\quad -x^{\frac{1}{2}} \delta {\mathcal {L}}_\eta ( \tfrac{\delta }{2} )\big )^{2m} \frac{\mathrm{d}x}{x} + O\big ((\log X)^{-\frac{1}{2}} (K^{\frac{1}{2}}\varepsilon _0 \log (\delta ^{-1}+2))^{2m}\big )\\&\ll \big (K \varepsilon _0^2\delta ^{\frac{1}{2}} (\log (\delta ^{-1}+2))^{\frac{3}{2}}\big )^m+(\log X)^{-\frac{1}{2}} \big (K^{\frac{1}{2}}\varepsilon _0 \log (\delta ^{-1}+2)\big )^{2m}, \end{aligned}$$

where $K>0$ is absolute and where we have bounded the part of the integral with $x\leqslant \exp ((\log X)^{\frac{1}{2}})$ using the uniform bound in Lemma 2.1. Recalling that $\delta = (\log _3 X )^3/(\log _2 X)^2$, for $\varepsilon _0^2 \delta ^{-\frac{1}{2}} (\log (\delta ^{-1}+2))^{\frac{1}{2}} \leqslant m \leqslant \varepsilon _0 \delta ^{-\frac{1}{2}} (\log (\delta ^{-1}+2))^{\frac{1}{2}}$, we have that $(\log X)^{- \frac{1}{m-1}} \leqslant \delta \leqslant \delta _0$, and hence Theorem 1.2 implies the lower bound

$$\begin{aligned} M_n(X,\delta ;\eta ,\Phi )&\geqslant (1+O(\varepsilon _0^2)) \mu _{2m} \big ( \tfrac{2}{3}\delta \log (\delta ^{-1}+2)\big )^m \\&\geqslant (2\pi )^{\frac{1}{2}} (1+O(\varepsilon _0^2)) \big ( \tfrac{2m}{3\mathrm{e}}\delta \log (\delta ^{-1}+2)\big )^m. \end{aligned}$$

When $\varepsilon _0$ is small enough, we obtain a contradiction as soon as the range

$$\begin{aligned} \varepsilon _0^2 K (\tfrac{3}{2} \mathrm{e}+\varepsilon _0) \delta ^{-\frac{1}{2}} (\log (\delta ^{-1}+2))^{\frac{1}{2}} \leqslant m\leqslant \varepsilon _0 \delta ^{-\frac{1}{2}} (\log (\delta ^{-1}+2))^{\frac{1}{2}} \end{aligned}$$

contains an integer; this is clearly the case when $\varepsilon _0$ is small enough and X is large enough. The proof of the first statement follows. The proof of the second is similar.   $\square$