Analytic continuation of multiple Hurwitz zeta functions

2008 ◽  
Vol 145 (3) ◽  
pp. 605-617 ◽  
Author(s):  
JAMES P. KELLIHER ◽  
RIAD MASRI
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Image Position

AbstractWe use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta function to $\mathbb{C}^{d}$, to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of $\zeta_{d}(s;\theta)$.

A Generalization of Epstein Zeta Functions

1949 ◽  
Vol 1 (4) ◽  
pp. 320-327 ◽  
Author(s):  
S. Minakshisundaram
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
General Class ◽  
Zeta Functions ◽  
Epstein Zeta Function ◽  
Image Position

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a1… ak are real and n1, n2, … nk run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein [2, 3]. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.

scholarly journals Relations among the Riemann Zeta and Hurwitz Zeta Functions, as Well as Their Products

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 754 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Formal Proof ◽  
Hurwitz Zeta Function ◽  
Mean Square ◽  
Novel Approach ◽  
Starting Point ◽  
Lindelöf Hypothesis ◽  
Riemann Zeta ◽  
The Mean

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large α behaviour of the modified Hurwitz zeta function ζ 1 ( s , α ) , s ∈ C , α ∈ ( 0 , ∞ ) , which is also presented here.

Inequalities for the Hurwitz zeta function

2000 ◽  
Vol 130 (6) ◽  
pp. 1227-1236 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Real Numbers ◽  
Positive Real ◽  
Image Position

Let be the Hurwitz zeta function. Furthermore, let p > 1 and α ≠ 0 be real numbers and n ≥ 2 be an integer. We determine the best possible constants a(p, α, n), A(p, α, n), b(p, n) and B(p, n) such that the inequalities and hold for all positive real numbers x1,…,xn.

scholarly journals ON APPROXIMATION OF ANALYTIC FUNCTIONS BY PERIODIC HURWITZ ZETA-FUNCTIONS

2018 ◽  
Vol 24 (1) ◽  
pp. 20-33 ◽  
Author(s):  
Darius Siaučiūnas ◽  
Violeta Franckevič ◽  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Sequence ◽  
Hurwitz Zeta Function ◽  
Complex Numbers ◽  
Closed Set

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental orrational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).

p-Adic Eigen-Eunctions for Kubert Distributions

1983 ◽  
Vol 35 (4) ◽  
pp. 674-686
Author(s):  
Neal Koblitz
Keyword(s):  
Zeta Function ◽  
Dimensional Space ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
One Dimensional ◽  
Fixed Complex ◽  
Image Position ◽  
Analogous Theorem ◽  
Positive Integers

Functions onR(or onR/Z, orQ/Z, or the interval (0,1)) which satisfy the identity1.1for positive integersmand fixed complexs,appear in several branches of mathematics (see [8], p. 65-68). They have recently been studied systematically by Kubert [6] and Milnor [12]. Milnor showed that for each complexsthere is a one-dimensional space of even functions and a one-dimensional space of odd functions which satisfy (1.1). These functions can be expressed in terms of either the Hurwitz partial zeta-function or the polylogarithm functions.My purpose is to prove an analogous theorem forp-adic functions. Thep-adic analog is slightly more general; it allows for a Dirichlet characterχ0(m) in front ofms–lin (1.1). The functions satisfying (1.1) turn out to bep-adic “partial DirichletL-functions”, functions of twop-adic variables (x, s) and one character variableχ0which specialize to partial zeta-functions whenχ0is trivial and to Kubota-LeopoldtL-functions whenx= 0.

Monotonicity Properties of the Hurwitz Zeta Function

2005 ◽  
Vol 48 (3) ◽  
pp. 333-339 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Real Numbers ◽  
Monotonicity Properties ◽  
Image Position

AbstractLetbe the Hurwitz zeta function and letwhereα, β> 1 anda,b> 0 are real numbers. We prove: (i) The functionQis decreasing on (0, ∞) iffαa−βb≥ max(a−b, 0). (ii)Qis increasing on (0, ∞) iffαa−βb≤ min(a−b, 0). An application of part (i) reveals that for allx> 0 the functions⟼ [(s− 1)ζ(s,x)]1/(s−1)is decreasing on (1, ∞). This settles a conjecture of Bastien and Rogalski.

scholarly journals Real zeros of Hurwitz–Lerch zeta and Hurwitz–Lerch type of Euler–Zagier double zeta functions

2015 ◽  
Vol 160 (1) ◽  
pp. 39-50 ◽  
Author(s):  
TAKASHI NAKAMURA
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Real Zeros ◽  
Double Zeta ◽  
Lerch Zeta Function

AbstractLet 0 < a ⩽ 1, s, z ∈ ${\mathbb{C}}$ and 0 < |z| ⩽ 1. Then the Hurwitz–Lerch zeta function is defined by Φ(s, a, z) ≔ ∑∞n = 0zn(n + a)− s when σ ≔ ℜ(s) > 1. In this paper, we show that the Hurwitz zeta function ζ(σ, a) ≔ Φ(σ, a, 1) does not vanish for all 0 < σ < 1 if and only if a ⩾ 1/2. Moreover, we prove that Φ(σ, a, z) ≠ 0 for all 0 < σ < 1 and 0 < a ⩽ 1 when z ≠ 1. Real zeros of Hurwitz–Lerch type of Euler–Zagier double zeta functions are studied as well.

scholarly journals Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Fundamental Constants ◽  
Closed Formula ◽  
Definite Integrals ◽  
Logarithmic Functions

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

scholarly journals REMARKS ON THE MIXED JOINT UNIVERSALITY FOR A CLASS OF ZETA FUNCTIONS

2016 ◽  
Vol 95 (2) ◽  
pp. 187-198 ◽  
Author(s):  
ROMA KAČINSKAITĖ ◽  
KOHJI MATSUMOTO
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Functional Independence ◽  
The Other ◽  
Hurwitz Zeta Function ◽  
Euler Product ◽  
Joint Universality

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when $\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

scholarly journals A note on the 2 k-th mean value of the Hurwitz zeta function

1999 ◽  
Vol 60 (3) ◽  
pp. 403-405 ◽  
Author(s):  
A. Kumchev
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Error Term ◽  
Mean Value ◽  
Substantial Improvement ◽  
Hurwitz Zeta Function ◽  
Image Position

Consider the error term in the asymptotic formulaIn this note we obtain δ(k) ≍ 1/(k6 log2k) which, for large values of k, presents a substantial improvement over the previously known result .

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