Transcendental Values of Some Dirichlet Series

Abstract

There is a large collection of Dirichlet series defined purely arithmetically that have been conjectured to have analytic continuation and functional equations. Deligne [38] has formulated a far-reaching conjecture regarding the special values of these series at special points in the complex plane and one would like to know if these special values are transcendental numbers or not. The most notable example is the L-function attached to an elliptic curve and the Birch and Swinnerton-Dyer conjecture. In a lecture at the Stony Brook conference on number theory in the summer of 1969, Sarvadaman Chowla posed the following question. Does there exist a rational-valued arithmetic function f, periodic with prime period p such that

$$\displaystyle{\sum _{n=1}^{\infty }{f(n) \over n} }$$

converges and equals zero? In 1973, Baker, Birch and Wirsing ([10], see also [29], [31] and [101]) answered this question in the following theorem: