Limit values of Eisenstein series and multiple cotangent functions

We show a limit formula for Eisenstein series by using the theory of a multiple cotangent function. The value is expressed simply via the Bernoulli number.     

Moments of the Riemann zeta function and Eisenstein series—II

The fourth moment of the Riemann zeta function and the second moment of the L-function of a Maass cusp form are studied using a construction of Epstein, Hafner and Sarnak.     

Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

Eisenstein series and theta functions to the septic base

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η⁴(7τ)/η⁴(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series.     

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

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions. Received: 17 Apri1 2007     

The joint universality theorem for a pair of Hurwitz zeta functions

In this paper we investigate the joint functional distribution for a pair of Hurwitz zeta functions ζ(s,αj) (j=1,2) in the case that real transcendental numbers α₁ and α₂ satisfy α₂∈Q(α₁). Especially we establish the joint universality theorem for these zeta functions.     

Addison-type series representation for the Stieltjes constants

The Stieltjes constants γk(a) appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about its only pole at s=1. We generalize a technique of Addison for the Euler constant γ=γ₀(1) to show its application to finding series representations for these constants. Other generalizations of representations of γ are given.     

An efficient algorithm for the Hurwitz zeta and related functions

A simple class of algorithms for the efficient computation of the Hurwitz zeta and related special functions is given. The algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision. A number of extensions as well as numerical examples are briefly described. The algorithms are easy to implement and compete with Euler–Maclaurin summation-based methods.     

On some series representations of the Hurwitz zeta function

A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants.     

Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.     

The Rogers-Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ₁, we provide a new proof of two Ramanujan's identities for the Rogers-Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers-Ramanujan continued fraction.     

Poles of Eisenstein series on quaternion groups

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.     

Simple zeros and discrete moments of the derivative of the Riemann zeta-function

We prove unconditional upper bounds for the second and fourth discrete moment of the first derivative of the zeta-function at its simple zeros on the critical line.     

A note on the Selberg class (degrees 0 and 1)

We give a classification of the Dirichlet polynomials in the Selberg class as entire quotients of Dirichlet series with periodic coefficients and the Riemann zeta-function. Received: 2 November 2001     

A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

Universality of the Riemann zeta-function

In 1975, S.M. Voronin proved the universality of the Riemann zeta-function ζ(s). This means that every non-vanishing analytic function can be approximated uniformly on compact subsets of the critical strip by shifts ζ(s+iτ). In the paper, we consider the functions F(ζ(s)) which are universal in the Voronin sense.     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

Asymptotic behavior of some power series with ζ-functions in the coefficients

The asymptotic behaviors of four closely related special cases of power series with -functions in the coefficients have appeared in the literature. A function which involves a real parameter is introduced; the parameter can be specialized to produce all of these special cases. The asymptotic behavior is obtained for this function for all real values of the parameter.