Related papers: Dynamical zeta functions and Kummer congruences
Multiple zeta functions of Arakawa-Kaneko and Euler-Zagier types are known as generalizations of the Riemann zeta function. In 2018, Kaneko and Tsumura proved that the multiple zeta functions of Arakawa-Kaneko type can be expressed as a…
In this paper we obtain the first mean-value theorems for the function $Z(t)$ on some disconnected sets. Next, we obtain a geometric law that controls chaotic behavior of the graph of the function $Z(t)$. This paper is the English version…
We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set and continuous maps in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le…
In this paper, we study graph-theoretic analogies of the Mertens' theorems by using basic properties of the Ihara zeta-function. One of our results is a refinement of a special case of the dynamical system Mertens' second theorem due to…
We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable $s$ in some right-half…
The present essay aims at investigating whether and how far an algebraic analysis of the Zeta Function and of the Riemann Hypothesis can be carried out. Of course the well-established properties of the Zeta Function, explored in depth in…
Following suggestions of T. H. Koornwinder, we give a new proof of Kummer's theorem involving Zeilberger's algorithm, the WZ method and asymptotic estimates. In the first section, we recall a classical proof given by L. J. Slater. The…
Using a cohomological characterization of the consistent and the covariant Lorentz and gauge anomalies, derived from the complexification of the relevant algebras, we study in $d=2$ the definition of the Weyl determinant for a non-abelian…
A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…
We prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz-Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on…
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise…
We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector…
A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist--Meurman--type congruences for the universal Bernoulli polynomials that are related…
The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n…
We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share…
The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment…
We introduce "puzzles of quasi-finite type" which are the counterparts of our subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of combinatorial puzzles as defined in complex dynamics. We are able to analyze these…
The special uniformity of zeta functions claims that pure non-abelian zeta functions coincide with group zeta functions associated to the special linear groups. Naturally associated are three aspects, namely, the analytic, arithmetic, and…
We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ…
Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…