scholarly journals Integral representations of the Riemann zeta function for odd-integer arguments

2002 ◽  
Vol 142 (2) ◽  
pp. 435-439 ◽  
Author(s):  
Djurdje Cvijović ◽  
Jacek Klinowski
Keyword(s):  
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 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 .

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals On The Tails of the Exponential Series

1994 ◽  
Vol 37 (2) ◽  
pp. 278-286 ◽  
Author(s):  
C. Yalçin Yildirim
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Partial Sums ◽  
Maclaurin Series ◽  
Exponential Series ◽  
Asymptotic Estimation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.

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