ASYMPTOTICS FOR HIGHER DERIVATIVES OF THE LERCH ZETA-FUNCTION: APPLICATIONS TO THE FORMULAE OF KUMMER, LERCH AND GAUSS (Analytic Number Theory and Related Topics)

Abstract

Let s be a complex variables, z a complex parameter, and a and λ real parameters with a > 0, and write e(s) = e2πis. The Lerch zeta-function φ(s, a, λ) is defined by the Dirichlet series Σ∞ l=0 e(λl)(a + l)−s (Res > 1), and its meromorphic continuation over the whole s-plane; this reduces to the Hurwitz zeta-function ζ(s, a) if λ is an integer, and further to the Riemann zeta-function ζ(s) = ζ(s, 1). Note that the domain of the parameter a can be extended through the procedure in [13]. Let φ(m)(s, z, λ) = (∂/∂s)mφ(s, z, λ) for m = 0, 1, 2, . . . denote any derivative. The aim of this paper is to show that complete asymptotic expansions exist for φ(m)(s, a + z, λ) (m = 0, 1, . . .) when both z → 0 and z → ∞ through | arg z| < π (Theorems 1 and 2), together with the explicit expressions of their remainders (Corollaries 1.1 and 2.2); these can be applied to deduce the classical Fourier series expansions of the log-gamma function log Γ(s) (Corollary 2.3) and the di-gamma function ψ(s) = (Γ'/Γ)(s) (Corollary 2.4) both for 0 < s < 1, due to Kummer and Lerch, respectively, as well as to deduce the celebrated closed form evaluation of ψ(r) at any rational point r with 0 < r < 1 (Corollary 2.5), due to Gauß. Our results in Theorems 1 and 2 further lead us to define and study a generalization of Deninger's Rm-function (Corollaries 1.4–1.6 and 2.6–2.9), which was first introduced by Deninger [3] for extending the log-gamma function into higher orders. The detailed proofs of our results in the present paper will appear, among other things, in the forthcoming article [21].