scholarly journals A Minimum Problem for the Epstein Zeta-Function

1953 ◽  
Vol 1 (4) ◽  
pp. 149-158 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Hexagonal Lattice ◽  
Mean Value ◽  
Minimum Problem ◽  
Constant Multiple ◽  
Packing And Covering ◽  
Closest Packing ◽  
Epstein Zeta Function ◽  
The Mean

In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of the Epstein zeta-function Z(s) associated with the lattice, taken at the point s=. Because of the connexion with the problems of closest packing and covering it seemed likely that the minimum value of Z() would be attained for the hexagonal lattice; it is the purpose of this paper to prove this and to extend the result to other real values of the variable s.

The mean value of the zeta-function on σ=1

2011 ◽  
Vol 26 (2) ◽  
pp. 209-227 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Mean Value ◽  
The Mean

scholarly journals On Riemann zeta-function and allied questions-II

1995 ◽  
Vol Volume 18 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Value ◽  
Dirichlet Polynomials ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience In this paper, two conjectures on the mean value of Dirichlet polynomials are given and are shown to imply good lower bound for $\int_H^{T+H}\vert\zeta(\frac{1}{2}+it)^k\vert^2\,dt$, uniform in $k$ and independent of $T$.

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 The mean-value of the Artin L-series and its derivative of a cubic field

1980 ◽  
Vol 21 (1) ◽  
pp. 9-18 ◽  
Author(s):  
Lenard Weinstein
Keyword(s):  
Entire Function ◽  
Zeta Function ◽  
Mean Value ◽  
Dedekind Zeta Function ◽  
Cubic Field ◽  
The Mean

Let K be a non-abelian cubic field of discriminant D, and ζK(s) its Dedekind zeta-function. Set ψ(s) = ζk(s)/ζ(s). Then it is known that ψ(s) is the Artin L-series associated with the field K. It is also known that ψ(s) is an entire function of order 1.

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