The Riemann zeta-function and moment conjectures from Random Matrix Theory

✍ Scribed by Jörn Steuding

• Book ID: 111493022
• Publisher: SP Versita
• Year: 2009
• Tongue: English
• Weight: 529 KB
• Volume: 59
• Category: Article
• ISSN: 0139-9918
• DOI: 10.2478/s12175-009-0129-0

✦ Synopsis

Abstract

On the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line, $$
\frac{1}
{T}\int\limits_0^T {|\zeta (\tfrac{1}
{2} + it)|^{2k} dt} and \frac{1}
{{N(T)}}\sum\limits_{0 < \gamma \leqslant {\rm T}} {|\zeta (\tfrac{1}
{2} + i(\gamma + \tfrac{\alpha }
{L}))|^{2k} }
$$, by Conrey, Keating et al. and Hughes, respectively. These conjectures are known to be true only for a few values of k and, even under assumption of the Riemann hypothesis, estimates of the expected order of magnitude are only proved for a limited range of k. We put the discrete moment for k = 1, 2 in relation with the corresponding continuous moment for the derivative of Hardy’s Z-function. This leads to upper bounds for the discrete moments which are off the predicted order by a factor of log T.