New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

The joint value-distribution of the Riemann zeta function and Hurwitz zeta functions

In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007.     

A footnote to mean square value of DirichletL-series

Following appropriate use of approximate functional equation for Hurwitz Zeta function, we obtain upper bounds for } Here fors = σ + it, L(s,x) denotes DirichletL-series for character x(modq). In particular, we obtain S(1/2 +it) ≪q logqt + t^5/8 q^−1/8, which is an improvement in the range q |t| < q^11/7, on hitherto best known result. This incidentally gives S(1/2+ it)≪ q log3q for |t|q^9/5.     

Derivatives of the Hurwitz Zeta function for rational arguments

The functional equation for the Hurwitz Zeta function ζ(s,a) is used to obtain formulas for derivatives of ζ(s,a) at negative odd s and rational a. For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function.     

Values of the Legendre chi and Hurwitz zeta functions at rational arguments

We show that the Hurwitz zeta function, , and the Legendre chi function, , defined by

and

respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.

     

An efficient algorithm for accelerating the convergence of oscillatory series,useful for computing the polylogarithm and Hurwitz zeta functions

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li _s (z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z ²/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.      

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

Another discrete Fourier transform pairs associated with the Lipschitz-Lerch zeta function

It is demonstrated that the alternating Lipschitz-Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function.     

Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions: III

The main object of this paper is the mean square I _h (s) of higher derivatives of Hurwitz zeta functions (s, ). We shall prove asymptotic formulas for I _h (1/2 + it) as t + with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for I _h (1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for I _h (1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product (u, )(v, ) with the independent complex variables u and v.     

Rational series for multiple zeta and log gamma functions

Text

We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-known properties of these functions. As corollaries many special values of these transcendental functions are expressed as series of higher order Bernoulli numbers.

Video

For a video summary of this paper, please click here or visit http://youtu.be/2i5PQiueW_8.     

Easy proofs of Riemann's functional equation for and of Lipschitz summation

We present a new, simple proof, based upon Poisson summation, of the Lipschitz summation formula. A conceptually easy corollary is the functional relation for the Hurwitz zeta function. As a direct consequence we obtain a short, motivated proof of Riemann's functional equation for .

     

Remarks on the universality of the periodic zeta function

We study the universality of a Dirichlet series with periodic coefficients. This property is proved in the case of multiplicative coefficients, and in the general case we establish universality in a certain set of analytic functions related to a probability distribution.     

Explicit Approximate Functional Equations for Various Classes of Dirichlet Series

We give explicit approximate functional equations for various classes of Dirichlet series, such as Lerch zeta function or Dirichlet L-series.Dedicated to Jean-Louis Nicolas on the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—11M06, 11M35, 11Y60     

On Some Integrals Involving the Hurwitz Zeta Function: Part 2

We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin(q), ln (q) and the polygamma functions. Many of the results are most conveniently formulated in terms of a family of functions A _k(q) := k(1 – k, q), k , and a family of polygamma functions of negative order, whose properties we study in some detail.     

On Some Integrals Involving the Hurwitz Zeta Function: Part 1

We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, ln (q) and ln sin(q).     

Series Acceleration Formulas for Dirichlet Series with Periodic Coefficients

Series acceleration formulas are obtained for Dirichlet series with periodic coefficients. Special cases include Ramanujan's formula for the values of the Riemann zeta function at the odd positive integers exceeding two, and related formulas for values of Dirichlet L-series and the Lerch zeta function.     

A limit theorem for the Hurwitz zeta function with an algebraic irrational parameter

A limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for the Hurwitz zeta function with an algebraic irrational parameter is obtained. Bibliography 6 titles. To the memory of Professor Yu. V. Linnik Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 125–134.     

Remark on a double-inequality for the Riemann zeta function

Let ζ be the Riemann zeta function and δ(x)=1/(2^x-1). For all x>0 we have

(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),