scholarly journals Shifted quadratic Zeta series

2004 ◽  
Vol 2004 (67) ◽  
pp. 3631-3652
Author(s):  
Anthony Sofo
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Point Of View ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is well known that the Riemann Zeta functionς(p)=∑n=1∞1/npcan be represented in closed form forpan even integer. We will define a shifted quadratic Zeta series as∑n=1∞1/(4n2−α2)p. In this paper, we will determine closed-form representations of shifted quadratic Zeta series from a recursion point of view using the Riemann Zeta function. We will also determine closed-form representations of alternating sign shifted quadratic Zeta series.

scholarly journals On the values of the Riemann zeta-function at rational arguments

2001 ◽  
Vol Volume 24 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
M Yoshimoto
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Companion Paper ◽  
Hurwitz Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Form Evaluation ◽  
Rational Arguments

International audience In a companion paper, ``On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', we obtained a closed form evaluation of Ramanujan's type of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator even and numerator odd), which was in turn a vast generalization of D. Klusch's and M. Katsurada's generalization of Ramanujan's formula. In this paper we shall continue our pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation thereof at all rational arguments, with no restriction to the form of the rationals, in the critical strip. This is a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-values.

Riemann Hypothesis from the Physicist’s Point of View

2020 ◽  
Vol 8 (3) ◽  
pp. 01-10
Author(s):  
Yuriy Zayko
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Point Of View ◽  
Distribution Of Primes ◽  
Computation Model ◽  
The Riemann Zeta Function ◽  
Real Argument ◽  
Riemann Zeta ◽  
Artificial Intelligence Systems

This article presents an attempt to comprehend the evolution of the ideas underlying the physical approach to the proof of one of the problems of the century - the Riemann hypothesis regarding the location of non-trivial zeros of the Riemann zeta function. Various formulations of this hypothesis are presented, which make it possible to clarify its connection with the distribution of primes in the set of natural numbers. A brief overview of the main directions of this approach is given. The probable cause of their failures is indicated - the solution of the problem within the framework of the classical Turing paradigm. A successful proof of the Riemann hypothesis based on the use of a relativistic computation model that allows one to overcome the Turing barrier is presented. This model has been previously applied to solve another problem not computable on the classical Turing machine - the calculation of the sums of divergent series for the Riemann zeta function of the real argument. The possibility of using relativistic computing for the development of artificial intelligence systems is noted.

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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