The pair correlation of zeros of the Riemann zeta function and distribution of primes (Q5929492)

In 1972 \textit{H. L. Montgomery} [Analytic number theory, Proc. Sympos. Pure Math. 24, 181-193 (1973; Zbl 0268.10023)] introduced the function \[ G(T,\xi) = \sum_{0< \gamma_1 , \gamma_2 \leq T}e(\xi(\gamma_1-\gamma_2))w(\gamma_1-\gamma_2), \] where \(w(u)= 4/(4+u^2)\), \(e(u)= e^{2\pi iu}\), and the sum is over pairs of imaginary parts of zeros \(\rho= \beta +i\gamma\) of the Riemann zeta-function. (Here we have changed Montgomery's notation slightly to agree with the paper being reviewed.) Assuming the Riemann hypothesis (RH), Montgomery evaluated \(G(T,\xi)\) asymptotically for \( |\alpha|\leq 1\), where \(\xi = \alpha {\log T\over 2 \pi}\), and he also conjectured that \(G(T,\xi) \sim {1\over 2\pi}T\log T\) for \(1\leq |\alpha|\leq M\) and any bounded number \(M\). This conjecture is referred to as the strong pair correlation conjecture, and as shown in [\textit{D. A. Goldston} and \textit{H. L. Montgomery}, Prog. Math. 70, 183-203 (1987; Zbl 0629.10032)] is equivalent under RH to an asymptotic formula for the second moment for primes in short intervals. In the present paper the authors are concerned with consequences for primes on assuming a pair correlation conjecture with a stronger error term. They formulate this conjecture in a somewhat different form than Montgomery based on work of \textit{Z. Rudnick} and \textit{P. Sarnak} [Duke Math. J. 81, 269-322 (1996; Zbl 0866.11050)] who proved an \(n\)-level correlation generalization of Montgomery's asymptotic formula in the range \(|\alpha|\leq 1\). Using this conjecture the authors first prove that assuming the RH one can obtain the strong pair correlation conjecture in the form \[ G(T,\xi) = \min\left(1, {2\pi \xi\over \log T}\right) {T\over 2\pi}\log T +e^{-4\pi \xi}{T\over 2\pi}\log^2 T + O(T) , \] uniformly for all \(\xi >0\); here the second term is only significant for \(\alpha\) close to zero and is not important in what follows. The error term here is what one would expect to be best possible, and one might go further and conjecture that there is a second order term \(c{T\over 2\pi}\) with \(c= -(1+\log 2 \pi)\) and a smaller error term, see for example \textit{D. Goldston} [Glasg. Math. J. 32, 285-297 (1990; Zbl 0719.11065)]. There is, however, a problem with assuming the conjecture holds uniformly in \(\xi\), since by Dirichlet's theorem one can find large values of \(\xi = \xi(T)\) for which all of the numbers \(e(\xi (\gamma_1 -\gamma_2))\), \(0<\gamma_1,\gamma_2\leq T\), are as close to 1 as we wish, which will violate the conjecture. Thus one should modify the conjecture in the paper to avoid this problem, probably by restricting \(\xi \ll \log T\) to match Montgomery's conjecture. In applying the stronger version of Montgomery's conjecture to primes, the authors use a method of \textit{D. R. Heath-Brown} [Acta Arith. 41, 85-99 (1982; Zbl 0485.10032)] where the pair correlation conjecture is used to bound \[ S(T,\xi) = \sum_{0<\gamma \leq T}e(\gamma \xi), \] which with the explicit formula can then be used to obtain information about primes. Heath-Brown showed that assuming RH and Montgomery's conjecture \(S(T,\xi) \ll T \log^{1\over 2} T\) for \(\log T \ll \xi \ll \log T\), and using the stronger form of the conjecture the authors obtain the improved bound \(\ll T\log ^{1\over 4}T\) in the range \( \log T\ll \xi \). From this they prove assuming RH that (1) \(\psi(x) - x \ll x^{1\over 2}\log^{5\over 4}x\), where \(\psi\) is the familiar Chebyshev function, (2) \(p_{n+1} - p_{n} \ll {p_n}^{1\over 2}\log^{1\over 4} p_n\), where \(p_n\) is the \(n\)th prime, and (3) \[ \liminf_{n\to \infty}{p_{n+1} - p_{n}\over \log^{2\over 3}p_n }< \infty . \] The result (1) is an improved error in the prime number theorem, but here the proof requires one to apply the conjecture in a range of \(\xi\) far larger than Montgomery's range. There is no problem with regard to the range of the conjecture in (2) and (3), where one only needs to assume the conjecture in a neighborhood of \(\alpha=2\) for (2) and \(\alpha =1\) for (3). It should be mentioned that it is possible to prove a stronger result than (3) with the same conjectures. The authors' result was obtained from a theorem of Heath-Brown, which in particular showed that Montgomery's conjecture in a neighborhood of \(\alpha =1\) implies \[ \liminf_{n\to \infty}{p_{n+1} - p_{n}\over \log p_n }=0 . \] Heath-Brown's theorem was not designed to take advantage of an error term smaller than \(O(T\log^{2\over 3}T)\) in the pair correlation conjecture, but in \textit{D. Goldston} [Thesis, U. C. Berkeley (1981)] the condition on the size of the error term was removed, and then Heath-Brown's theorem implies that assuming the conjecture with error term \(O(T)\) then \[ \liminf_{n\to \infty}(p_{n+1} - p_{n}) \leq C \] for an absolute constant \(C\). It might also be possible that (2) can be improved to show that with the same conjectures \(p_{n+1} - p_{n} \ll {p_n}^{1\over 2}\).