Masami Yoshimoto

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.

Farey series and the Riemann hypothesis

This is a direct continuation of Part II of this series of papers and we shall not only improve (numerically) all our former results in Part II, but also prove new theorems, Theorem 4 on the function f(t) = log Γ(t+λ) and Theorem 5 for k-th power moments, k = 2,...,7, by elaborating our previous arguments.

Ramanujan’s Formula and Modular Forms

In the theory of zeta-functions, which are defined wherever there are defined norms or substitutes thereof, the ingredients — modular relations, functional equations, incomplete gamma series, and the like — are placed like nodes on the woofs. Some of them are woven by warps as Hecke theory or Lavrik’s theory. The former connects the modular relation to the functional equation, thus making it possible to go to and from between the more orderly world of automorphic forms and the less orderly one of zeta-functions while the latter relates functional equations and incomplete gamma series in the same vein, the idea originating from Riemann. We have found a warp stitching all of these nodes-ingredients, enabling us to warp from one node to another as well as providing us with a guiding principle to locate the exact position and direction of research, a guiding thread to give a clear picture of the whole scene through opaque mist of complexity. We shall illustrate the principle by examples of various zeta-functions satisfying Hecke’s functional equation, i.e. the one with a single gamma factor, in which category many of the important zeta-functions are contained, notably, the Riemann zeta-, Dirichlet L-, Epstein zeta-, the automorphic zeta-functions, etc. In particular, we shall be concerned with the automorphic zeta-functions, the zeta functions arising from automorphic forms, evaluating their special values and obtaining incomplete gamma series.

On Rapidly Convergent Series Expressions for Zeta- and L-Values, and Log Sine Integrals

Regarding the rapidly convergent series expansion for special values of ζ- and L-functions for integer points, there are two approaches.One approach starts from Eulers 1772 formula for ζ(3) and culminates in Srivastavas very recent results via many intermediate results, and the other is due to Wiltons investigation, which was shown by us (Aeq. Math.59, 2000, 1–19) to be a consequence of Ramanujans work (Collected Papers of Srinivasa Ramanujan, CUP 1927, reprint Chelsea, 1962, pp. 163–168).More recently, Katsurada (Acta Arith.90, 1999, 79–89.) has generalized all existing formulas into a rather wide framework of Dirichlet L-functions.Our purpose is to show that even the most general Katsuradas formulas are easy consequences of our fundamental summation formulas for the series with Hurwitz zeta-function coefficients.We give a three-line proof of Katsuradas main theorem, and also we make some remarks on the recent paper of Bradley (The Ramanujan J.3, 1999, 159–173).

On Bessel series expressions for some lattice sums: II

In part I (Kanemitsu S et al 2003 J. Northwest University) we have made explicit use of the Mellin–Barnes integrals to prove the Chowla–Selberg-type Bessel series expressions for zeta-functions associated with lattice structures. In this paper we shall make implicit use of Mellin–Barnes integrals, as embedded in our theory of modular relations and functional equations, to reveal relationships between the structure of Madelung constants of the NaCl and CsCl lattices. Namely, we shall elucidate the relation between the structures of the NaCl lattice and those of the CsCl lattice, so to speak using the symmetry of the zeta-function, i.e. using their functional equations. Thus we shall emphasize the symmetry properties of the zeta-functions, restoring the Schlomilch series and Hardys theory of K-Bessel functions, to prove the functional equations, and then to prove the recurrence relations for the lattice zeta-functions.

Crystal Symmetry Viewed as Zeta Symmetry II

In this paper, we continue our previous investigations on applications of the Epstein zeta-functions. We shall mostly state the results for the lattice zeta-functions, which can be immediately translated into those for the corresponding Epstein zeta-functions. We shall take up the generalized Chowla–Selberg (integral) formula and state many concrete special cases of this formula.

On some slowly convergent series involving the Hurwitz zeta-function

We shall extract the essence of the Adamchik-Srivastava generating function method (Analysis (Munich) 18 (1998) 131) by proving the most far-reaching Ramanujan-Yoshimoto formula and by showing that some of the results stated in Srivastava and Choi (Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001) are simple consequences of the above-mentioned formula.

Some Aspects of the Modular Relation

This paper is a companion to the forthcoming paper [19] and exhibits various manifestations of the modular relation, equivalent to the functional equation. We shall give a somewhat new proof of the functinal equation for the Hurwitz-Lerch Dirichlet L-functions in §1, elucidation of Chan’s result relating the functional equation to the q-series (or vice versa) in §2, while §3 and §4 are devoted to elucidate the location of the partial fraction expansion of the coth (cot, respectively) in the modular relation framework.