Haruo Tsukada

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.

SOME NUMBER THEORETIC APPLICATIONS OF A GENERAL MODULAR RELATION

We state a form of the modular relation in which the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman.

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.

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.