The mean value theorem for the Riemann zeta‐function

Mathematika ◽  
1978 ◽  
Vol 25 (2) ◽  
pp. 177-184 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Mean

scholarly journals Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1

1978 ◽  
Vol Volume 1 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Positive Constant ◽  
Riemann Zeta Function ◽  
Nonnegative Integer ◽  
Mean Value ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience The present paper is concerned with $\Omega$-estimates of the quantity $$(1/H)\int_{T}^{T+H}\vert(d^m/ds^m)\zeta^k(\frac{1}{2}+it)\vert dt$$ where $k$ is a positive number (not necessarily an integer), $m$ a nonnegative integer, and $(\log T)^{\delta}\leq H \leq T$, where $\delta$ is a small positive constant. The main theorems are stated for Dirichlet series satisfying certain conditions and the corollaries concerning the zeta function illustrate quite well the scope and interest of the results. %It is proved that if $2k\geq1$ and $T\geq T_0(\delta)$, then $$(1/H)\int_{T}^{T+H}\vert \zeta(\frac{1}{2}+it)\vert^{2k}dt > (\log H)^{k^2}(\log\log H)^{-C}$$ and $$(1/H)\int_{T}^{T+H} \vert\zeta'(\frac{1}{2}+it)\vert dt > (\log H)^{5/4}(\log\log H)^{-C},$$ where $C$ is a constant depending only on $\delta$.

scholarly journals On mean value results for the Riemann zeta-function in short intervals.

2009 ◽  
Vol Volume 32 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Value ◽  
Upper And Lower Bounds ◽  
Mean Values ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience We discuss the mean values of the Riemann zeta-function $\zeta(s)$, and analyze upper and lower bounds for $$\int_T^{T+H} \vert\zeta(\frac{1}{2}+it)\vert^{2k}\,dt~~~~~~(k\in\mathbb{N}~{\rm fixed,}~1<\!\!< H \leq T).$$ In particular, the author's new upper bound for the above integral under the Riemann hypothesis is presented.

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