scholarly journals AN UPPER BOUND FOR DISCRETE MOMENTS OF THE DERIVATIVE OF THE RIEMANN ZETA‐FUNCTION

Mathematika ◽  
2020 ◽  
Vol 66 (2) ◽  
pp. 475-497
Author(s):  
Scott Kirila
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Upper Bound ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Moments

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

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
Jörn Steuding
Keyword(s):  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Limited Range ◽  
Theory Model ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Moments

AbstractOn 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.

scholarly journals On a problem of Ivi\'c.

2000 ◽  
Vol Volume 23 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Upper Bound ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Implicit Constant ◽  
International Audience ◽  
Riemann Zeta

International audience Let $\gamma$ denote the imaginary parts of the nontrivial zeros of the Riemann zeta-function $\zeta(s)$. For sufficiently large $T$ and $\varepsilon>0$, Ivi\'c proved that $\sum_{T<\gamma\leq2T} \vert\zeta(\frac{1}{2}+i\gamma)\vert^2 <\!\!\!<_{\varepsilon} (T(\log T)^2\log\log T)^{3/2+\varepsilon},$ where the implicit constant depends only on $\varepsilon$. In this paper, this result is improved by (i) replacing $\vert\zeta(\frac{1}{2}+i\gamma)\vert^2$ by $\max\vert\zeta(s)\vert^2$, where the maximum is taken over all $s=\sigma+it$ in the rectangle $\frac{1}{2}-A/\log T\leq\sigma\leq2,\, \vert t-\gamma\vert\leq B(\log\log T)/\log T$ with some fixed positive constants $A, B,$ and (ii) replacing the upper bound by $T(\log T)^2\log\log T$. The method of proof differs completely from Ivi\'c's approach.

scholarly journals Hybrid Moments of the Riemann Zeta-Function

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Upper Bound ◽  
Mellin Transform ◽  
Critical Line ◽  
Mean Square ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The “hybrid” moments ∫T2Tζ1/2+itk∫t-Gt+Gζ1/2+ixldxmdt  Tε≪G=GT≪T of the Riemann zeta-function ζs on the critical line Res=1/2 are studied. The expected upper bound for the above expression is Oε(T1+εGm). This is shown to be true for certain specific values of k,l,m∈N, and the explicitly determined range of G=G(T;k,l,m). The application to a mean square bound for the Mellin transform function of ζ1/2+ix4 is given.

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