scholarly journals ANALOGUES OF A TRANSFORMATION FORMULA OF RAMANUJAN

2011 ◽  
Vol 07 (05) ◽  
pp. 1151-1172 ◽  
Author(s):  
ATUL DIXIT
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Zeta Functions ◽  
Transformation Formula ◽  
Lost Notebook ◽  
Riemann Zeta ◽  
Limiting Case ◽  
Special Case

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma function and Riemann zeta function present in his Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular-type relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A. P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.

Zeta Functions and the Log Behaviour of Combinatorial Sequences

2015 ◽  
Vol 58 (3) ◽  
pp. 637-651 ◽  
Author(s):  
William Y. C. Chen ◽  
Jeremy J. F. Guo ◽  
Larry X. W. Wang
Keyword(s):  
Probability Theory ◽  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Bernoulli Number ◽  
Hölder’S Inequality ◽  
Zeta Functions ◽  
Bell Number ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractIn this paper, we use the Riemann zeta functionζ(x) and the Bessel zeta functionζμ(x) to study the log behaviour of combinatorial sequences. We prove thatζ(x) is log-convex forx> 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n≥ 1 is log-convex, whereBnis thenth Bernoulli number. We introduce the functionθ(x) = (2ζ(x)Γ(x + 1))1/x, whereΓ(x)is the gamma function, and we show that logθ(x) is strictly increasing forx≥ 6. This confirms a conjecture of Sun stating that the sequenceis strictly increasing. Amdeberhanet al. defined the numbersan(μ)= 22n+1(n+ 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence{an(μ)}n≥1is log-convex forμ= 0 andμ= 1. By proving thatζμ(x)is log-convex forx >1 andμ >-1, we show that the sequence{an(≥)}n>1 is log-convex for anyμ >- 1. We introduce another functionθμ,(x)involvingζμ(x)and the gamma functionΓ(x)and we show that logθμ(x)is strictly increasing forx >8e(μ+ 2)2. This implies thatBased on Dobinski’s formula, we prove thatwhereBnis thenth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property ofand Holder’s inequality in probability theory.

scholarly journals Fundamental Domains of Gamma and Zeta Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Cabiria Andreian Cazacu ◽  
Dorin Ghisa
Keyword(s):  
Conformal Mapping ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Riemann Surfaces ◽  
Zeta Functions ◽  
Branched Covering ◽  
Fundamental Domains ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Color Visualization

Branched covering Riemann surfaces(ℂ,f)are studied, wherefis the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of cover transformations is revealed. In order to find fundamental domains, preimages of the real axis are taken and a thorough study of their geometry is performed. The technique of simultaneous continuation, introduced by the authors in previous papers, is used for this purpose. Color visualization of the conformal mapping of the complex plane by these functions is used for a better understanding of the theory. A version of this paper containing colored images can be found in arXiv at Andrian Cazacu and Ghisa.

scholarly journals Convexity Properties and Inequalities Concerning the (p,k)-Gamma function

2017 ◽  
Author(s):  
Kwara Nantomah
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Convexity Properties ◽  
Riemann Zeta

In this paper, some convexity properties and some inequalities for the (p,k)-analogue of the Gamma function, Гp,k(x) are given. In particular, a (p,k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p,k)-analogue of the Riemann zeta function, ζp,k(x) is introduced and some associated inequalities are derived. The established results provide the (p,k)-generalizations of some known results concerning the classical Gamma function.

scholarly journals Joint universality of the Riemann zeta-function and Lerch zeta-functions

2013 ◽  
Vol 18 (3) ◽  
pp. 314-326
Author(s):  
Antanas Laurinčikas ◽  
Renata Macaitienė˙
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Joint Universality Theorem

In the paper, we prove a joint universality theorem for the Riemann zeta-function and a collection of Lerch zeta-functions with parameters algebraically independent over the field of rational numbers.

scholarly journals UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

2021 ◽  
Vol 26 (1) ◽  
pp. 82-93
Author(s):  
Aidas Balčiūnas ◽  
Violeta Franckevič ◽  
Virginija Garbaliauskienė ◽  
Renata Macaitienė ◽  
Audronė Rimkevičienė
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Pair Correlation ◽  
Zeta Functions ◽  
Weak Form ◽  
Cusp Forms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

The Consequence of the Analytic Continuity of Zeta Function Subject to an Additional Term and a Justification of the Location of the Non-Trivial Zeros

2020 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Additional Term ◽  
Riemann Zeta

We review two main results of Riemann Zeta function; the analytic continuity and the first functional equation by the means of Gamma function and Hankel contour. We observe that an additional term is considered in both results. We justify the non-trivial location of Zeta non-trivial zeros subject to an approximation.

Simultaneous Polynomial Approximations of the Lerch Function

2009 ◽  
Vol 61 (6) ◽  
pp. 1341-1356 ◽  
Author(s):  
Tanguy Rivoal
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Zeta Functions ◽  
Detailed Comparison ◽  
Padé Approximants ◽  
Bivariate Polynomial ◽  
Integer Points ◽  
Polynomial Approximations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite–Padé approximants either of the polylogarithms or ofHurwitz zeta functions. In the former case, we recover known results, while in the latter the results are new and generalise some recent works of Beukers and Prévost. Finally, we make a detailed comparison of our work with Beukers’. Such constructions are useful in the arithmetical study of the values of the Riemann zeta function at integer points and of the Kubota–Leopold p-adic zeta function.

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