Related papers: Riemann zeta functions for Krull monoids
In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined…
The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library,…
We give a conditional lower bound on the number of non-trivial simple zeros for the Dedekind zeta function $\zeta_{K}(s)$, where $K$ is a quadratic number field. The conditional result is given by assuming a Lindel\"of on average (in the…
The aim of this work is to improve some elementary results regarding both the Deuring-Phenomenon and the Heilbronn-Phenomenon. We will give better estimates regarding both the influence of zeros of the Riemann zeta function on the…
We propose a regularization technique and apply it to the Euler product of zeta functions, mainly of the Riemann zeta function, to make unknown some clear. In this paper that is the first part of the trilogy, we try to demonstrate the…
We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and p-adic…
In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give…
After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of…
In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number…
We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…
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…
Some aspects of adelic generalized functions, as linear continuous functionals on the space of Schwartz-Bruhat functions, are considered. The importance of adelic generalized functions in adelic quantum mechanics is demonstrated. In…
In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our…
This is an expanded version. We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a…
We develop a general method for computing the homological Euler characteristic of finite index subgroups G of GL_m(O_K) where O_K is the ring of integers in a number field K. With this method we find, that for large, explicitly computed…
We give a conjecture for the moments of the Dedekind zeta function of a Galois extension via the hybrid product method. The moments of the product of primes are evaluated using the Montgomery-Vaughan mean value theorem whilst for the…
We have proposed a regularization technique and apply it to the Euler product of zeta functions in the part one. In this paper that is the second part of the trilogy, we give another evidence to demonstrate the Riemann hypotheses by using…
We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials…
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 establish precise relations between Euler systems that are respectively associated to a $p$-adic representation $T$ and to its Kummer dual $T^*(1)$. Upon appropriate specialization of this general result, we are able to deduce the…