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.

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 Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

scholarly journals Explicit formulas for local factors in the Euler products for Eisenstein series

1989 ◽  
Vol 113 ◽  
pp. 37-87 ◽  
Author(s):  
Paul Feit
Keyword(s):  
Dirichlet Series ◽  
Rational Function ◽  
Eisenstein Series ◽  
Fourier Coefficients ◽  
Simple Condition ◽  
Explicit Formulas ◽  
Local Factors ◽  
Euler Products ◽  
Infinite Sums

Our objective is to prove that certain Dirichlet series (in our variable q−s), which are defined by infinite sums, can be expressed as a product of an explicit rational function in q−s times an unknown polynomial M in q−s Moreover we show that M(q−s) is 1 if a simple condition is met. The Dirichlet series appear in the Euler products of Fourier coefficients for Eisenstein series. The series discussed below generalize the functions α0(N, q−s) used by Shimura in [12], and the theorem is an extension of Kitaoka’s result [5].

scholarly journals Відносне зростання рядів Діріхле з різними абсцисами абсолютної збіжності

2020 ◽  
Vol 72 (11) ◽  
pp. 1535-1543
Author(s):  
O. M. Mulyava ◽  
M. M. Sheremeta
Keyword(s):  
Dirichlet Series ◽  
Lower Order ◽  
Absolute Convergence ◽  
Entire Dirichlet Series ◽  
Function Inverse ◽  
Increasing Function

УДК 517.537.72 We study the growth of a Dirichlet series with zero abscissa of absolute convergence with respect to the entire Dirichlet series by using the generalized quantities of order  and lower order  where   is the function inverse to and is a positive increasing function growing to  

scholarly journals Behavior of Digital Sequences Through Exotic Numeration Systems

10.37236/6581 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Julien Leroy ◽  
Michel Rigo ◽  
Manon Stipulanti
Keyword(s):  
Number Theory ◽  
Linear Algebra ◽  
Analytic Number Theory ◽  
Binomial Coefficients ◽  
Periodic Fluctuation ◽  
Sum Of Digits ◽  
Analytic Number ◽  
Summatory Function ◽  
Sum Of Digits Function ◽  
Numeration Systems

Many digital functions studied in the literature, e.g., the summatory function of the base-$k$ sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply it on two examples motivated by the study of generalized Pascal triangles and binomial coefficients of words.

scholarly journals On the growth of a class of Dirichlet series absolutely convergent in half-plane

2017 ◽  
Vol 9 (1) ◽  
pp. 63-71
Author(s):  
L.V. Kulyavetc' ◽  
O.M. Mulyava
Keyword(s):  
Dirichlet Series ◽  
Half Plane ◽  
Absolute Convergence ◽  
Increasing Function

In terms of generalized orders it is investigated a relation between the growth of a Dirichlet series $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with the abscissa of asolute convergence $A\in (-\infty,+\infty)$ and the growth of Dirichlet series $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$, $1\le j\le 2$, with the same abscissa of absolute convergence, if the coefficients $a_n$ are connected with the coefficients $a_{n,j}$ by correlation $$ \beta\left(\frac{\lambda_n}{\ln\,\left(|a_n|e^{A\lambda_n}\right)}\right)=(1+o(1)) \prod\limits_{j=1}^{m}\beta\left(\frac{\lambda_n} {\ln\,\left(|a_{n,j}|e^{A\lambda_n}\right)}\right)^{\omega_j},\  n\to\infty, $$ where $\omega_j>0$ $(1\le j\le m)$, $\sum\limits_{j=1}^{m}\omega_j=1$, and $\alpha$ is a positive slowly increasing function on $[x_0, +\infty)$.

On Joint Distribution of General Dirichlet Series

2003 ◽  
Vol 8 (2) ◽  
pp. 27-39 ◽  
Author(s):  
J. Genys ◽  
A. Laurinčikas
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Dirichlet Series ◽  
Joint Distribution ◽  
Meromorphic Functions ◽  
General Dirichlet Series ◽  
Joint Limit Theorem ◽  
Space Of Meromorphic Functions ◽  
Weaker Conditions ◽  
Joint Limit

In the paper a joint limit theorem in the sense of the weak convergence in the space of meromorphic functions for general Dirichlet series is proved under weaker conditions as in [1].

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