Pair correlation of the zeros of the Riemann zeta function in longer ranges

Abstract

In this paper, we extend the result of Fujii on the second moment of S(t + h) − S(t) to longer range of h under the Riemann Hypothesis and an quantitative form of the Twin Prime Conjecture.

On the number variance of zeta zeros and a conjecture of Berry

“…In Section 1.3, we give three different formulations of these results, stated as Theorems 1.3.1-1.3.3. In Section 1.4, we show how our results give a conditional proof of Berry's conjecture in the non-universal regime assuming RH and a conjecture of Chan [5] for the pair correlation of zeta zeros in longer ranges (which examines how often normalized gaps between zeros can be close to a fixed non-zero value). Roughly, pair correlation studies the distribution of gap sizes localized near zero with respect to the average spacing, whereas our new results require information about the distribution of gap sizes localized near other points.…”

“…Our result relies on finer information from both the primes and the zeros of 𝜁(𝑠). In particular, we require a variation of Montgomery's function 𝐹(𝛼) introduced by Chan [5] in his study of the pair correlation of zeros in longer ranges. We define…”

“…In contrast, the function $$F_\Delta (\alpha )$$ is no longer positive, nor real, nor even; however, it satisfies the symmetry relations $$$$\begin{equation} \overline{F_\Delta (\alpha )}=F_\Delta (-\alpha )=F_{-\Delta }(\alpha ). \end{equation}$$$$Combining the methods of Chan [5, Theorem 1.1] with Goldston–Montgomery [16], it can be shown that $$$$\begin{equation} F_\Delta (\alpha ) = T^{-2\alpha }\log T + \alpha \, w{\left(\Delta \right)}T^{- i\alpha \Delta } + O{\left(\frac{1}{\sqrt {\log T}}\right)} + O(T^{-2\alpha })+ O_\varepsilon {\left(\frac{(\Delta +1)\,T^{-\alpha {\left(\tfrac{1}{2}-\varepsilon \right)}}}{\log T}\right)}, \end{equation}$$$$uniformly for $$0\leqslant \alpha \leqslant 1$$ and small $$\varepsilon >0$$.…”

“…Conceivably part (b) of Conjecture 1.4.1 continues to hold for δ in a much longer range. We require the following generalization of the strong form of Montgomery's Pair Correlation conjecture due to Chan [5, Conjecture 1.1]. Conjecture For $$|\alpha |\geqslant 1$$ and $$\Delta =o(\log ^{\frac{1}{3}}T)$$, we have $$$$\begin{equation*} F_\Delta (\alpha ) = T^{- i \alpha \Delta }\,w{\left(\Delta \right)}{\left(1+o(1)\right)}, \end{equation*}$$$$uniformly for α in compact intervals as $$T\rightarrow \infty$$.…”

“…As we will see, the restriction $$\Delta =o(\log ^2 T)$$ in Theorems 1.3.1–1.3.3, corresponding to $$\delta =o(\log ^3 T)$$, arises naturally in two different places. It first arises from the last error term in the formula (1.17) for $$F_\Delta (\alpha )$$ from [5, Theorem 1.1] (see Lemmas 3.1.1 and 3.3.2), and it again arises from estimating a sum over primes in Lemma 4.2.1. In the following sections, we attempt to state each lemma in the largest possible range of Δ to clarify where these restrictions appear.…”