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). (1.6), the Lerch zeta-function (1.4) is less well known, the existing monograph [ 36 ] notwithstanding. Recently there has been a new representation-theoretic interpretation of the Lerch zeta-function, cf. e.g. [ 35 ]. In the last few decades, the most fundamental and influential works related to the Lerch zeta-function are [ 16 ], [ 43 ], [ 47 ], and [ 71 ], which are partly incorporated in [ 10 ]. We shall describe these toward the end of this section. In the paper [ 9 ] the ubiquity of the Lerch zeta-function, especially the monologarithm (cid:96) 1 ( x ) (1.22) of the complex exponential argument, has been pursued.