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ON THE PAIR CORRELATION OF ZEROS OF THE RIEMANN ZETA-FUNCTION

Published online by Cambridge University Press:  01 January 2000

D. A. GOLDSTON
Affiliation:
Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192, USAgoldston@jupiter.sjsu.edu
S. M. GONEK
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, USAgonek@math.rochester.edu
A. E. ÖZLÜK
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA and Research Institute of Mathematics, Orono ozluk@gauss.umemat.maine.edusnyder@gauss.umemat.maine.edu
C. SNYDER
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA and Research Institute of Mathematics, Orono ozluk@gauss.umemat.maine.edusnyder@gauss.umemat.maine.edu
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Abstract

To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function $$ F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), $$ where $\alpha$ is real and $T\ge 2$, $\gamma$ and $\gamma '$ 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 $F(\alpha)$ when $|\alpha|\le 1$, and made the conjecture that $F(\alpha) = 1 + o(1)$ as $T\to \infty$ for any bounded $\alpha$ with $|\alpha |\ge 1$. 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 $\epsilon >0$ we have $$ F(\alpha) \ge {3\over 2} - |\alpha| - \epsilon ,$$ uniformly for $1\le |\alpha| \le \frac32 -2\epsilon $ and all $T \ge T_0(\epsilon)$.

1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.

Type
Research Article
Copyright
2000 London Mathematical Society

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