scholarly journals CERTAIN INTEGRAL REPRESENTATIONS FOR THE RIEMANN ZETA FUNCTION ζ(s) AT POSITIVE INTEGER ARGUMENT

2013 ◽  
Vol 35 (4) ◽  
pp. 639-645
Author(s):  
Junesang Choi
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integral Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals A Probabilistic Proof for Representations of the Riemann Zeta Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 369
Author(s):  
Jiamei Liu ◽  
Yuxia Huang ◽  
Chuancun Yin
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Probabilistic Approach ◽  
Recurrence Formula ◽  
Integral Representations ◽  
The Riemann Zeta Function ◽  
Probabilistic Proof ◽  
Riemann Zeta ◽  
Positive Integers

In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.

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 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.

scholarly journals On certain sums over the non-trivial zeta zeroes

2010 ◽  
Vol 466 (2124) ◽  
pp. 3679-3692 ◽  
Author(s):  
Mark W. Coffey
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Asymptotic Form ◽  
Integral Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We study coefficients b n , expressible as sums over the Li/Keiper constants λ j , that contain information on the Riemann xi function. We present a number of relations for and representations of b n . These include the expression of b n as a sum over non-trivial zeroes of the Riemann zeta function, as well as integral representations. Conditional on the Riemann hypothesis, we provide the asymptotic form of .

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