HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Journal articles

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

Abstract : 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 $\delta$.
Document type :
Journal articles
Complete list of metadata

Cited literature [5 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01103723
Contributor : Ariane Rolland Connect in order to contact the contributor
Submitted on : Thursday, January 15, 2015 - 11:53:24 AM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
Long-term archiving on: : Thursday, April 16, 2015 - 10:30:25 AM

File

1Article1.pdf
Explicit agreement for this submission

Identifiers

Collections

Citation

K Ramachandra. Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1978, Volume 1 - 1978, pp.1-15. ⟨10.46298/hrj.1978.87⟩. ⟨hal-01103723⟩

Share

Metrics

Record views

124

Files downloads

525