scholarly journals On the meromorphic continuation of Beatty Zeta-functions and Sturmian Dirichlet series

2019 ◽  
Vol 194 ◽  
pp. 303-318 ◽  
Author(s):  
Athanasios Sourmelidis
Keyword(s):  
Dirichlet Series ◽  
Zeta Functions ◽  
Meromorphic Continuation

Scalar Products of Certain Hecke L-Series and Moments of Weighted Norm-Counting Functions

1985 ◽  
Vol 28 (3) ◽  
pp. 272-279 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Dirichlet Series ◽  
Finite Groups ◽  
Tensor Products ◽  
Meromorphic Continuation ◽  
Weighted Norm ◽  
Mellin Transformation ◽  
Counting Functions ◽  
Representations Of Finite Groups ◽  
Scalar Products

AbstractWe consider Dirichlet series R(s), constructed by taking scalar products of Hecke L-series with ray-class characters. Using a theorem of G. W. Mackey on tensor products of representations of finite groups we show that R(s) has a meromorphic continuation into Re(s) > 1/2 (obtained by more sophisticated methods in [l]-[5]); we then obtain estimates for the growth of R(s) on vertical lines. Via the Mellin transformation we deduce asymptotics for various weighted moment sums involving ideals of given ray-class and norm, in one or several fields simultaneously.

ON A FORMULA FOR THE REGULARIZED DETERMINANT OF ZETA FUNCTIONS WITH APPLICATION TO SOME DIRICHLET SERIES

2020 ◽  
Vol 71 (3) ◽  
pp. 843-865
Author(s):  
Mounir Hajli
Keyword(s):  
Dirichlet Series ◽  
Large Class ◽  
Zeta Functions ◽  
Special Values

Abstract In this paper, we study a large class of zeta functions. We evaluate explicitly the special values of these zeta functions and the associated derivatives at $0$. As an application, we recover several results on the zeta functions defined by two polynomials already obtained in the literature.

scholarly journals Meromorphic continuation of Koba-Nielsen string amplitudes

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
M. Bocardo-Gaspar ◽  
Willem Veys ◽  
W. A. Zúñiga-Galindo
Keyword(s):  
Characteristic Zero ◽  
Zeta Functions ◽  
Open String ◽  
Local Fields ◽  
Meromorphic Continuation ◽  
Partition Functions ◽  
Interesting Connection ◽  
Bona Fide ◽  
Local Zeta Functions ◽  
String Amplitudes

Abstract In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.

scholarly journals Similar Sublattices of Planar Lattices

2011 ◽  
Vol 63 (6) ◽  
pp. 1220-1537 ◽  
Author(s):  
Michael Baake ◽  
Rudolf Scharlau ◽  
Peter Zeiner
Keyword(s):  
Dirichlet Series ◽  
Generating Functions ◽  
Quadratic Field ◽  
Zeta Functions ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Planar Lattice ◽  
Planar Lattices

AbstractThe similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

scholarly journals Joint universality of periodic zeta-functions with multiplicative coefficients

2020 ◽  
Vol 25 (5) ◽  
Author(s):  
Antanas Laurinčikas ◽  
Monika Tekorė
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Coefficients ◽  
Joint Universality

The periodic zeta-function is defined by the ordinary Dirichlet series with periodic coefficients. In the paper, joint universality theorems on the approximation of a collection of analytic functions by nonlinear shifts of periodic zeta-functions with multiplicative coefficients are obtained. These theorems do not use any independence hypotheses on the coefficients of zeta-functions.

scholarly journals Dirichlet series associated to sum-of-digits functions

2021 ◽  
pp. 1-22
Author(s):  
Corey Everlove
Keyword(s):  
Dirichlet Series ◽  
Meromorphic Functions ◽  
Absolute Convergence ◽  
Meromorphic Continuation ◽  
Explicit Formulas ◽  
Sum Of Digits ◽  
Summatory Function ◽  
Continuous Interpolation

We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-[Formula: see text] digits of the integer [Formula: see text], and [Formula: see text], where [Formula: see text] is the summatory function of [Formula: see text]. We show that [Formula: see text] and [Formula: see text] have analytic continuations to the plane [Formula: see text] as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions [Formula: see text] and [Formula: see text] to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.

scholarly journals Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

2018 ◽  
Vol 14 (08) ◽  
pp. 2317-2331
Author(s):  
Marcus du Sautoy
Keyword(s):  
Elliptic Curves ◽  
Zeta Functions ◽  
Symmetric Power ◽  
Meromorphic Continuation ◽  
Natural Boundary ◽  
Integer Polynomials ◽  
Analytic Behavior ◽  
Euler Products ◽  
Natural Boundaries

We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.

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