THE BOUNDARY LERCH ZETA-FUNCTION AND SHORT CHARACTER SUMS À LA Y. YAMAMOTO

Abstract

As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185-195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann's fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628-634). We may thus refer to this as the ‘Fourier series-boundary q-series', and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275-289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori, the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).