Path integral representation of the Riemann zeta function

1995 ◽  
Vol 45 (1) ◽  
pp. 89-90
Author(s):  
P. Babinec
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Path Integral ◽  
Riemann Zeta Function ◽  
Path Integral Representation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals A new derivation of $\zeta(-r)=-\frac{B_{r+1}}{r+1}$

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Master Theorem

In this note, we give a new derivation for the fact that $\zeta(-r)=-\frac{B_{r+1}}{r+1}$ where $\zeta(s)$ represents the Riemann zeta function, and $B_{r}$ represents the Bernoulli numbers. Our proof uses the well-known explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind, and the Ramanujan's master theorem to obtain an integral representation for 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 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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