Related papers: A fully new path to prove Riemann Hypothesis
We introduce a differential topological proof and an analytical proof of Riemann hypothesis according to the saddle point method because Riemann calculated the integral representation of zeta function on the critical line by this method.…
We give an informal survey of the historical development of computations related to prime number distribution and zeros of the Riemann zeta function.
A formula of Doetsch ({\em Math. Zeitschr.} {\bf 42}, 263 (1937)) is generalized and used to numerically invert the one-sided Laplace transform ${\hat C}(\beta)$. The necessary input is only the values of ${\hat C}(\beta)$ on the positive…
We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…
Geometric generalized Mittag-Leffler distributions having the Laplace transform $\frac{1}{1+\beta\log(1+t^\alpha)},0<\alpha\le 2,\beta>0$ is introduced and its properties are discussed. Autoregressive processes with Mittag-Leffler and…
Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the…
In this paper we perform a detailed analysis of Riemann's hypothesis, dealing with the zeros of the analytically-extended zeta function. We use the functional equation $\zeta(s) = 2^{s}\pi^{s-1}\sin{(\displaystyle \pi…
This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $\psi(x)$. We shall be extensively using complex analytic…
This work is dedicated to the promotion of the results Hadamard, Landau E., Walvis A., Estarmann T and Paul R. Chernoff for pseudo zeta functions. The properties of zeta functions are studied, these properties can lead to new regularities…
In recent years, there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis. Most of these kind of contributions are…
A proof of the Riemann hypothesis using the reflection principle is presented.
In this article we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle. We start by considering the special case of the star graph with Neumann matching conditions at the center of…
The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions…
An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. The partial fraction expansion, p(s), of f(s) is obtained using the conjunction of the Riemann hypothesis and hypotheses advanced…
The Riemann zeta function and the distribution of its nontrivial zeros on the critical line remain central topics in analytic number theory and large-scale computation. This work develops a numerical framework that replaces classical…
This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…
We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the…
For the Riemann zeta-function, we introduce a function such that it is a characteristic function of an infinitely divisible distribution on the real line if and only if the Riemann Hypothesis is true.
Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros…
This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…