scholarly journals The Zeta and Related Functions: Recent Developments

2019 ◽  
Vol 3 (1) ◽  
pp. 329 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Zeta Function ◽  
Convergent Series ◽  
Creative Commons ◽  
Lerch Zeta Function ◽  
Leonhard Euler ◽  
Recent Developments ◽  
Expository Article ◽  
Riemann Zeta ◽  
Open Access Article ◽  
Generalized Zeta Function

The main object of this survey-cum-expository article is to present an overview of some recent developments involving the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta function ζ(s, a), and the Hurwitz-Lerch Zeta function Φ(z, s, a), which have their roots in the works of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707–1783) and the Russian mathematician, Christian Goldbach (1690–1764). We aim at considering the problems associated with the evaluations and representations of ζ(s) when s ∈ N \ {1}, N is the set of natural numbers, with emphasis upon several interesting classes of rapidly convergent series representations for ζ(2n+1) (n ∈ N). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux will also be provided for supporting their computational usefulness. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

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 Generating Function Identities for $\zeta(2n+2), \zeta(2n+3)$ via the WZ Method

10.37236/759 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Kh. Hessami Pilehrood ◽  
T. Hessami Pilehrood
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Convergent Series ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Wz Method

Using WZ-pairs we present simpler proofs of Koecher, Leshchiner and Bailey-Borwein-Bradley's identities for generating functions of the sequences $\{\zeta(2n+2)\}_{n\ge 0}$ and $\{\zeta(2n+3)\}_{n\ge 0}.$ By the same method, we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function.

Random matrices and number theory: some recent themes

2018 ◽  
Author(s):  
Jon P. Keating
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gaussian Random Fields ◽  
Extreme Value Statistics ◽  
Matrix Methods ◽  
Recent Developments ◽  
Riemann Zeta

The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.

scholarly journals Globally convergent series for the Riemann zeta function and the Dirichlet beta function

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Beta Function ◽  
Series Representation ◽  
Convergent Series ◽  
Globally Convergent ◽  
Series Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Dirichlet Beta Function

We derive the following globally convergent series for the Riemann zeta function and the Dirichlet beta function$$\zeta(s)=\frac{1}{2^{s}-2}\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}}\binom{2k+1}{k+1}\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}} \qquad \mbox{(where $s \neq 1+\frac{2\pi i n}{\ln 2}$)},$$$$\beta(s)=\frac{1}{4^{s}}\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\left(\left(\frac{3}{4}\right)^{(k+1)}-\left(\frac{1}{4}\right)^{(k+1)}\right)\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}}$$using a globally convergent series for the polylogarithm function, and integrals representing the Riemann zeta function and the Dirichlet beta function. To the best of our knowledge, these series representations are new. Additionally, we give another proof of Hasse's series representation for the Riemann zeta function.

scholarly journals Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II

1980 ◽  
Vol Volume 3 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Convergent Series ◽  
Mean Value ◽  
The Riemann Zeta Function ◽  
Dirichlet Polynomials ◽  
Riemann Zeta ◽  
The Mean ◽  
Real Truth

International audience This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{-C}$ represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate $2k$ is an integer. %This is of great interest, for little has been known on the mean value of $\vert\zeta(\frac{1}{2}+it)\vert^k$ for odd $k$, say $k=1$; for even $k$, see the book by E. C. Titchmarsh [The theory of the Riemann zeta function, Clarendon Press, Oxford, 1951, Theorem 7.19]. The proofs are based on applications of classical function-theoretic theorems, together with mean value theorems for Dirichlet polynomials or series. %In the case of the zeta function, the principle is to write $\vert\zeta(s)\vert^k=\vert\zeta(s)^{k/2}\vert^2$, where $\zeta(s)^{k/2}$ is related to a rapidly convergent series which is essentially a partial sum of the Dirichlet series of $\zeta(s)^{k/2}$, convergent in the half-plane $\sigma>1$.

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 .

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