Kohji Matsumoto

The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I

Abstract We consider general multiple zeta-functions of multi-variables, including both Barnes multiple zeta-functions and Euler–Zagier sums as special cases. We prove the meromorphic continuation to the whole space, asymptotic expansions, and upper bound estimates. These results are expected to have applications to some arithmetical L -functions (such as of Hecke and of Shintani). The method is based on the classical Mellin–Barnes integral formula.

Asymptotic series for double zeta, double gamma, and Hecke L -functions

Asymptotic expansions of the Barnes double zeta-function formula here and the double gamma-function Γ 2 (α, (1, w )), with respect to the parameter w , are proved. An application to Hecke L -functions of real quadratic fields is also discussed.

Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions

The purpose of the present article is to survey some mean value results obtained recently in zeta-function theory. We do not mention other important aspects of the theory of zeta-functions, such as the distribution of zeros, value-distribution, and applications to number theory. Some of them are probably treated in the articles of Professor Apostol and Professor Ramachandra in the present volume. Even in the mean value theory, we do not discuss many important recent topics. Those include: Recent progress in the theory of large values and fractional moments made by Heath-Brown [55] and the Indian school (Ramachandra, Balasubramanian, Sankaranarayanan and others, see Ramachandra [172]); mean values taken at the zeros or at the points near the zeros (Gonek [30] [31], Fujii [23]-[26] and others); the mean square of the product of the zeta-function and a Dirichlet polynomial (see Conrey-Ghosh-Gonek [19] and the papers quoted there). All of these three topics are closely connected with the distribution of zeros of zeta-functions, hence the full account of them would require too many pages. We will only discuss the theory of Titchmarsh series very briefly in Section 7. In the fourth power moment theory there have been remarkable developments which may be characterized by the use of the spectral theory of Maass wave forms. We mention this theory occasionally, but only in connection with the mean square problems. For the full details of this theory, see Chapters 4 and 5 of Ivić [68], Motohashi’s book [155], and Jutila’s series of papers. In the present article we only discuss the mean square theory of zeta-functions. This is a rather restricted topic, but still it is impossible to mention all the relevant results because the recent progress in this area is very big. The main tools appearing in this article are the approximate functional equations and Atkinson methods, emphasis are laid on the latter. The readers will find, however, that these two tools are not irrelevant (see Sections 4 and 6). Efforts are made to explain the mutual connections among various methods and results. In Section 1, we summarize the results on the mean square Iσ(T ) of the Riemann zeta-function, obtained by applying various approximate functional equations. In Sections 3 and 5, Iσ(T ) is studied ¿from the viewpoint of the method of Atkinson. The background of Atkinson’s method is the divisor problem, which is mentioned in Sections 2 and 6. Then, after a brief discussion on some short interval results in Section 7, we proceed to survey the results on more general zeta and L-functions. Sections 8, 9, 10 and 11 are devoted, respectively, to the mean square theory of Dedekind zeta-functions, L-functions attached to cusp forms, Dirichlet L-functions, and Hurwitz zeta and other related zeta-functions.

Functional equations for double zeta-functions

As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as as pecial case.

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

The main object of this paper is the mean square I h (s) of higher derivatives of Hurwitz zeta functions ζ(s, α). 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 Atkinsons dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

ZETA-FUNCTIONS OF ROOT SYSTEMS

In this paper, we introduce multi-variable zeta-functions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zeta-functions associated with Lie algebras which can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case of type Ar, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types Ar (r = 1, 2, 3) which include what is called Witten’s volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups.

Analytic Properties of Multiple Zeta-Functions in Several Variables

We report several recent results on analytic properties of multiple zeta-functions, mainly in several variables, such as the analytic continuation, the asymptotic behaviour, the location of singularities, and the recursive structure. Some results presented in this paper have never been published before.

Asymptotic probability measures of zeta-functions of algebraic number fields

Abstract We prove the existence of asymptotic probability measures connected with the value-distribution of the Dedekind zeta-function ζK(s) of an arbitrary algebraic number field K. This is a generalization of Bohr-Jessens classical result, which has shown the existence in the case of the Riemann zeta-function. We note that, to prove our theorem in the non-Galois case, we must take a way which is independent of the properties of convex plane curves. Also, we obtain a quantitative estimate in case K is Galois, which is an improvement of the authors former result.

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

AbstractFor 1/2<σ<1 fixed, letEσ(T) denote the error term in the asymptotic formula for