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 New series involving the zeta function

2001 ◽  
Vol 28 (7) ◽  
pp. 403-411 ◽  
Author(s):  
Wu Yun-Fei
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Double Gamma Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We evaluate sums of certain classes of new series involving the Riemann zeta function by using the theory of the double gamma function and a property of the gamma function. Relevant connections with various known results are also pointed out.

scholarly journals Some series involving the zeta function

1995 ◽  
Vol 51 (3) ◽  
pp. 383-393 ◽  
Author(s):  
Junesang Choi ◽  
H.M. Srivastava ◽  
J.R. Quine
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Double Gamma Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Determinants Of Laplacians

Lots of formulas for series of zeta function have been developed in many ways. We show how we can apply the theory of the double gamma function, which has recently been revived according to the study of determinants of Laplacians, to evaluate some series involving the Riemann zeta function.

Biased behavior of weighted Mertens sums

2019 ◽  
Vol 16 (03) ◽  
pp. 547-577
Author(s):  
Emre Alkan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Summation Formulas ◽  
The Riemann Zeta Function ◽  
Convexity Properties ◽  
Riemann Zeta ◽  
Derivatives Of

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at [Formula: see text].

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 Ramanujan's formula for the logarithmic derivative of the gamma function

1996 ◽  
Vol 120 (3) ◽  
pp. 391-401
Author(s):  
David Bradley
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Logarithmic Derivative ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Positive Integers ◽  
Euler's Constant

AbstractWe prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

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