Scalar product of L-functions with Grössencharacters: its meromorphic continuation and natural boundary.

Abstract

1. Introduction oo oo ooThe convolution Σ a n b"n~s of two Dirichlet series Σ a n n~* and Σ b n n~s is a n=l n=1 n-l classic object studied by many authors, especially in the context of the theory of automoφhic forms (we refer to [8], [30] and [28], or to the more recent work [27], [10], [18], [1], for example). In connection with the multidimensional arithmetic of E. Hecke, [9], [29], [11], Yu. V. Linnik suggested, [19], to consider the scalar product of Hecke's L-functions with Gr encharakters and asked whether this function can be analytically continued to the whole complex plane C. In 1971, P.K.J. Draxl, [4], proved that it has a meromorphic continuation to the half-plane C + = {,s|Re$>0}, and a year later O. M. Fomenko, [5], showed that for two quadratic fields this function can be continued to the whole complex plane C. A few years ago N. Kurokawa, [12], [13], [14], proved that the scalar product of Hecke's functions has a natural boundary C 0 = {slRe-^O} and cannot be continued to C~ = {5|Εβ5<0}, unless either all the fields, but possibly one of them, coincide with the ground field, or two of them are quadratic extensions and the others, if any, coincide with the ground field. It has been done under an additional assumption that all the Gr ssencharakters under consideration are of finite order, We show here that the restriction on the order of characters can be removed, if one assumes the validity of the Extended Riemann's Hypothesis for all the Hecke L-functions of algebraic number fields. Our treatment is elementary and self-contained, but in many places we follow [12] rather closely.