Related papers: Trace formulas applied to the Riemann $\zeta$-func…
This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product…
We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the…
In this paper, we give an approximate formula for the measure of extreme values for the logarithm of the Riemann zeta-function and its iterated integrals. The result recovers the unconditional best result for the $\Omega$-result of…
In this work we derive a functional equation in terms of the Hurwitz-Lerch zeta function along with definite integrals in terms of the incomplete gamma and Hurwitz-Lerch zeta functions. The method used in these derivations is contour…
We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.
A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the examined density are derived.
This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.
The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s}…
Spectra of real alternating operators seem to be quite interesting from the view point of explaining the Riemann Hypothesis for various zeta functions. Unfortunately we have not sufficient experiments concerning this theme. Necessary works…
We prove new relations on zeta function at even arguments and Dirichlet $L$ function at odd. The key idea is to make use of the Taylor series and partial fraction decomposition of cotangent and secant functions as we discuss in calculus and…
We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cram\'er, and also new representations for some arithmetical…
The inverse scattering transform is extended to investigate the Tzitz\'{e}ica equation. A set of sectionally analytic eigenfunctions and auxiliary eigenfunctions are introduced. We note that in this procedure, the auxiliary eigenfunctions…
Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include…
We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.
This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of…
This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties…
Renormdynamic equations of motion and their solutions are given. New equation for NBD distribution and Riemann zeta function invented. Explicit forms of the z-Scaling functions are constructed.
We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.
We discuss some aspects of the search for identities using computer algebra and symbolic methods. The focus is on so-called Apery-like formulae for special values of the Riemann Zeta function. Much work lays ahead in formally proving and…
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…