ON THE PAIR CORRELATION OF ZEROS OF THE RIEMANN ZETA-FUNCTION
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...
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| Published in: | Proceedings of the London Mathematical Society Vol. 80; no. 1; pp. 31 - 49 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cambridge University Press
01.01.2000
Oxford University Press |
| Subjects: | |
| ISSN: | 0024-6115, 1460-244X |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ArticleID:80.1.31 istex:6FB0D6EBD6DE3E7E811C223794E24F45A9AF1C45 ark:/67375/HXZ-0VJTJDWK-M |
| ISSN: | 0024-6115 1460-244X |
| DOI: | 10.1112/S0024611500012211 |