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

In: Mathematica Slovaca, vol. 59, no. 3
Jörn Steuding

## Details:

Year, pages: 2009, 323 - 338
Keywords:
Riemann zeta-function, discrete moments, nontrivial zeros
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 $2k$th continuous and discrete moments of the zeta-function on the critical line,
$${1\over T}\int0T \b\vertζ\b({\textstyle{1\over 2}+ \ii t}\bb)\bb\vert2k \d t \mbox{and} {1\over N(T)}∑0<γ≤ T \b\vertζ\b({\textstyle{1\over 2} + \ii \b(γ+{α\over L}\bb)}\bb)\bb\vert2k,$$
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 $log T$.