Explicit relations between pair correlation of zeros and primes in short intervals (Q442501)

Let \(\zeta(s)\) denote the Riemann zeta-function, and let \(\rho=\beta+i\gamma\) denote a nontrivial zero of \(\zeta(s)\). Montgomery's pair correlation function is defined as NEWLINE\[NEWLINE F(X,T) = \sum_{0<\gamma, \gamma'\leq T} X^{i(\gamma-\gamma')} w(\gamma-\gamma'), \quad w(u)=\frac{4}{4+u^2} NEWLINE\]NEWLINE where the sum runs over two sets of ordinates of zeros of \(\zeta(s)\). The paper under review explores the equivalence between the asymptotic behavior of the function \(F(X,T)\) and the distribution of primes in short intervals extending and improving results of \textit{T. H. Chan} [J. Lond. Math. Soc. (2) 68 (2003) 579--598; Zbl 1054.11047]. To state the main results, write NEWLINE\[NEWLINE F(X,T) = \frac{T}{2\pi} \left( \log \frac{T}{2\pi} - 1 \right) + R_F(X,T) NEWLINE\]NEWLINE and write NEWLINE\[NEWLINE \int_X^{2X} \left( \psi(x+\theta x)-\psi(x)-\theta x \right)^2 dx = \frac{3}{2} X^2 \theta \left( \log(1/\theta)+C\right) + R_J(X,\theta), NEWLINE\]NEWLINE where \(\psi(x)=\sum_{n\leq x} \Lambda(n)\) is the usual summatory function of von Mangoldt's function \(\Lambda(n)\) and \(C=1-\gamma_0-\log 2\pi\) (\(\gamma_0\) is Euler's constant). Assuming the Riemann hypothesis (RH), the authors investigate the relationship between the error terms \(R_F(X,T)\) and \(R_J(X,\theta)\) with \(X, T,\) and \(\theta\) in suitable ranges.NEWLINENEWLINEThey prove:NEWLINENEWLINE1. Assume RH and let \(0<\alpha <1\), \(\beta\geq 0\), and \(1\leq A_1 \leq A_2\). If NEWLINE\[NEWLINE R_F(X,T) \ll T^{1-\alpha} \log^{-\beta}T NEWLINE\]NEWLINE uniformly for \(X^{1/A_2}/\log^{(3+\beta)}X\leq T \leq X^{1/A_1}/\log^{(3+\beta)}X\), then NEWLINE\[NEWLINE R_J(X,\theta) \ll X^2 \theta^{1+\alpha/2} \log^{(1-\beta)/2}(1/\theta) NEWLINE\]NEWLINE uniformly for \(X^{-1/A_1(1+\alpha)} \leq \theta \leq X^{-1/A_2(1-\alpha)}\).NEWLINENEWLINE2. Assume RH and let \(0<\alpha <1\), \(\beta\geq 0\), \((\alpha,\beta)\neq (0,0)\), and \(1\leq A_1 \leq A_2\). If NEWLINE\[NEWLINE R_J(X,\theta) \ll X^2 \theta^{1+\alpha} \log^{-\beta}(1/\theta) NEWLINE\]NEWLINE uniformly for \(X^{-(\alpha/2+1)/A_1}/\log^{(3+\beta)}X \leq \theta \leq X^{-1/A_2}/\log^{4(\beta+2)/3}X\), then NEWLINE\[NEWLINE R_F(X,T) \ll T^{1-\alpha/(\alpha+3)} (\log T)^{(\alpha-\beta+2)/(\alpha+3)} NEWLINE\]NEWLINE uniformly for \(X^{(\alpha+3)/3A_2(1-\alpha)} \leq T \leq X^{(\alpha+3)/3A_1}\).NEWLINENEWLINEThe authors also discuss the relationship between these results and the integral NEWLINE\[NEWLINE \int_X^{2X} \left( \psi(x+h)-\psi(x)-h \right)^2 dx NEWLINE\]NEWLINE which they note is perhaps the ``more natural'' way to study primes in short intervals.