Complex Zeros of an Incomplete Riemann Zeta Function and of the Incomplete Gamma Function

1970 ◽  
Vol 24 (111) ◽  
pp. 679 ◽  
Author(s):  
K. S. Kolbig
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Incomplete Gamma Function ◽  
Riemann Zeta ◽  
Complex Zeros

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.

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.

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 A note on Hardy's theorem

2017 ◽  
Vol Volume 39 ◽  
Author(s):  
Usha K. Sangale
Keyword(s):  
Zeta Function ◽  
Simple Proof ◽  
Riemann Zeta Function ◽  
Hardy’S Theorem ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
Complex Zeros

International audience Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

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.

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.

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