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.

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 $\zeta_{d}(s;\theta)$.

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

scholarly journals Generalised Dirichlet series and Hecke's functional equation

1967 ◽  
Vol 15 (4) ◽  
pp. 309-313 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
General Class ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
Riemann Zeta ◽  
Image Position

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

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.

scholarly journals A lemma about the Epstein zeta-function

1964 ◽  
Vol 6 (4) ◽  
pp. 198-201 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Positive Definite ◽  
Simple Pole ◽  
Positive Definite Quadratic Form ◽  
Epstein Zeta Function ◽  
Image Position

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shapewith y > 0. We write (for s > 1)The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

scholarly journals On the values of the Epstein zeta function

1973 ◽  
Vol 14 (1) ◽  
pp. 1-12 ◽  
Author(s):  
John Roderick Smart
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Bernoulli Number ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
The Riemann Zeta Function ◽  
Epstein Zeta Function ◽  
Upper Half Plane ◽  
Riemann Zeta ◽  
Image Position

Let ζ(s) = σn-s(Res >1) denote the Riemann zeta function; then, as is well known,, whereBmdenotes themth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integerss = k> 2. Letbe a positive definite quadratic form andwhere the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]where γ is Euler's constant,is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formulaOn the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).

scholarly journals On a problem of Rankin about the Epstein zeta-function

1959 ◽  
Vol 4 (2) ◽  
pp. 73-80 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Special Form ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Epstein Zeta Function ◽  
Image Position

Letbe a positive definite quadratic form with determinant αβ−X2 = 1. A special form of this kind isWe consider the Epstein zeta-functionthe series converging for s > 1. For s ≥ 1·035 Rankin [1] proved the followingSTatement R.The sign of equality is needed only when h is equivalent to Q.

On a problem about the Epstein zeta-function

1964 ◽  
Vol 60 (4) ◽  
pp. 855-875 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Special Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Simple Pole ◽  
Epstein Zeta Function ◽  
Image Position

1. Letbe a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind isWe consider the Epstein zeta-functionthe series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.

scholarly journals RAMANUJAN SERIES UPSIDE-DOWN

2014 ◽  
Vol 97 (1) ◽  
pp. 78-106 ◽  
Author(s):  
JESUS GUILLERA ◽  
MATHEW ROGERS
Keyword(s):  
Zeta Function ◽  
Hypergeometric Functions ◽  
Zeta Functions ◽  
Epstein Zeta Function

AbstractWe prove that there is a correspondence between Ramanujan-type formulas for$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $and formulas for Dirichlet$L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

On the Riemann zeta-function

1932 ◽  
Vol 28 (3) ◽  
pp. 273-274 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Absolute Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfyingwhere A is an absolute constant.

Sign in / Sign up

Export Citation Format

Share Document