scholarly journals An integral involving the generalized zeta function

1990 ◽  
Vol 13 (3) ◽  
pp. 453-460 ◽  
Author(s):  
E. Elizalde ◽  
A. Romeo
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Surfaces ◽  
The Riemann Zeta Function ◽  
Special Integral ◽  
Riemann Zeta ◽  
Compact Riemann Surfaces ◽  
Determinants Of Laplacians ◽  
Generalized Zeta Function

A general value for∫abdtlogΓ(t), fora,bpositive reals, is derived in terms of the Hurwitzζfunction. That expression is checked for a previously known special integral, and the case whereais a positive integer andbis half an odd integer is considered. The result finds application in calculating the numerical value of the derivative of the Riemann zeta function at the point−1, a quantity that arises in the evaluation of determinants of Laplacians on compact Riemann surfaces.

scholarly journals Some series involving the zeta function

1995 ◽  
Vol 51 (3) ◽  
pp. 383-393 ◽  
Author(s):  
Junesang Choi ◽  
H.M. Srivastava ◽  
J.R. Quine
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Double Gamma Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Determinants Of Laplacians

Lots of formulas for series of zeta function have been developed in many ways. We show how we can apply the theory of the double gamma function, which has recently been revived according to the study of determinants of Laplacians, to evaluate some series involving the Riemann zeta function.

scholarly journals On a question of Ramachandra

1982 ◽  
Vol Volume 5 ◽  
Author(s):  
hugh L Montgomery
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Sigma 1 ◽  
International Audience ◽  
Riemann Zeta

International audience For each positive integer $k$, let $$a_k(n)=(\sum_p p^{-s})^k=\sum_{n=1}^{\infty} a_k(n)n^{-s},$$ where $\sigma={\rm Re}(s)>1$, and the sum on the left runs over all primes $p$. This paper is devoted to proving the following theorem: If $1/2<\sigma<1$, then $$\max_k(\sum_{n\leq N} a_k(n)^2n^{-2\sigma})^{1/2k}\approx (\log N)^{1-\sigma}/\log\log N$$ and $$(\sum_{n=1}^{\infty} a_k(n)^2n^{-2\sigma})^{1/2k} \approx k^{1-\sigma}/(\log k)^{\sigma}.$$ The constants implied by the $\approx$ sign may depend upon $\sigma$. This theorem has applications to the Riemann zeta function.

scholarly journals An integral representation for the Riemann zeta function on positive integer arguments

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Positive Integer ◽  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this brief note, we give an integral representation for the Riemann zeta function for positive integer arguments. To the best of our knowledge, the representation is new.

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