ON THE PAIR CORRELATION OF ZEROS OF THE RIEMANN ZETA-FUNCTION - GOLDSTON, D. A.; GONEK, S. M.; ÖZLÜK, A. E.; SNYDER, C. (Proceedings of the London Mathematical Society, 2000)
To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function $$ F(α) = F_T(α) = \left({T\over 2π}\log T\right)^{-1} \sum_{0<γ,γ ' \le T} T^{iα(γ -γ ')}w(γ-γ '), $$ where $α$ is real and $T\ge 2$, $γ$ and $γ '$ denote the imaginary parts of zeros of the Riemann zeta-function, and $w(u) = 4/(4 + u^2)$. Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for $F(α)$ when $|α|\le 1$, and made the conjecture that $F(α) = 1 + o(1)$ as $T→ ∞$ for any bounded $α$ with $|α |\ge 1$. In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions, then for any $ε >0$ we have $$ F(α) \ge {3\over 2} - |α| - ε ,$$ uniformly for $1\le |α| \le \frac32 -2ε $ and all $T \ge T_0(ε)$. 1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.