Related papers: Hypergeometric-like Representation of the Zeta-Fun…
The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. We study the structure of the twisted homology and cohomology…
Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
In this paper we prove the existence and uniqueness of a topological quantum field theory that incorporates, for all Riemann surfaces, the corresponding spaces of theta functions and the actions of the Heisenberg groups and modular groups…
In the paper, we introduce $q$-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of…
In this article, with a new approach, which is not discussed in the literature yet, the limit of the Riemann zeta function or Euler-Riemann zeta function is approximately explored by applying Dirichlet's rearrangement theorem for absolutely…
In this paper, we give an overview of the various general methods in computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo $p^m$ of the zeta function of a…
This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…
Based on $k$-gamma and $k$-digamma functions, we show four series expansions to the Furdui-type integral related to Riemann zeta function and hypergeometric function, and also present some new identities, series expansions and inequalities…
In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work "Remarques…
We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are…
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…
In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two…
We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity condition that we prove to be…
We introduce a new Tauberian framework through the theory of "regular arithmetic functions". This allows us to establish a characterization of the Riemann hypothesis by linking the floor function to the distribution of nontrivial zeros of…
A new expansion for integral powers of the hypergeometric function corresponding to a special case of the incomplete beta function is summarized, and consequences, including two new sums involving digamma (psi) functions are presented.
A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the $\zeta$-function approach is presented. The method produces finite quantities directly and finite scale-parametrized counterterms at…
In this article, we describe a new algorithm for the expansion of hypergeometric functions about half-integer parameters. The implementation of this algorithm for certain classes of hypergeometric functions in the already existing…
The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to…
The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to…
Two integral representations of q-analogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at…