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 An Alternative perspective to Riemann Hypothesis

2021 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Functional Equation ◽  
Integral Representation ◽  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Additional Term ◽  
Alternative Perspective ◽  
Riemann Zeta

We review some main results of Riemann Zeta function; the Integral representation 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 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 Generalised Dirichlet series and Hecke's functional equation

1967 ◽  
Vol 15 (4) ◽  
pp. 309-313 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
General Class ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
Riemann Zeta ◽  
Image Position

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

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.

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