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

hrj:87 - Hardy-Ramanujan Journal, January 1, 1978, Volume 1 - https://doi.org/10.46298/hrj.1978.87
Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1

Authors: K Ramachandra

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 $\delta$.


Volume: Volume 1
Published on: January 1, 1978
Submitted on: March 3, 2015
Keywords: Riemann zeta-function,Dirichlet series.,Ω-estimates,[MATH] Mathematics [math]


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