Related papers: Rationality of secant zeta values

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also…

We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.

We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving…

We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be…

Using elementary methods we find surprising connections between the values of the Riemann Zeta Function over integers and the fractional parts of rational powers, and a connection between the Riemann Zeta Function and the Prime Zeta…

Spencer Bloch and the author formulated a general conjecture (Tamagawa number conjecture) on the relation between values of zeta functions of motives and arithmetic groups associated to motives. We discuss this conjecture, and describe some…

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…

In this paper, by introducing a new operation in the vector space of Laurent series, the author derived explicit series for the values of $\zeta$-funtion at positive integers, where $\zeta$ denotes the Riemann zeta function. The values of…

We calculate zeta functions for certain orders of rank $3$ defined by standard integral table algebras and integral fusion rings that have irrational-valued irreducible characters. The calculations are obtained from explicit calculations of…

In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a…

In this paper, for any positive integer $\ell\geq2,$ we define $\ell$-generalized Fibonacci zeta function. We then study its analytic continuation to the whole complex plane $\mathbb{C}.$ Further, we compute a possible list of singularities…

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over $\mathbb Z$. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products…

It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations.

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…

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete…

The theory of Selberg zeta functions is generalized to higher rank spaces. Applications towards analytic torsion numbers are given.

We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. The…

A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\zeta_{k}$ and the method of the proof of the…

Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Maslanka, Coffey, Baez-Duarte, Voros and…