Joint value-distribution of L-functions and discrepancy of Hecke eigenvalues∗

Abstract

This paper consists of three parts. In the first part, we extend a classical result of Bohr and Courant about values of the Riemann zeta-function.We actually establish a joint denseness result about values of Dirichlet L-functions and automorphic L-functions for SL(2, ℤ) by using a weak version of Selberg’s orthogonality and a probabilistic limit theorem. The second part concerns the discrepancy D(N, f) of the Hecke eigenvalues λf (p) with respect to the Sato–Tate measure, where f is a holomorphic primitive form of SL(2, ℤ), for which the Sato–Tate conjecture was proved. We estimate the discrepancy D(N, f) toward Akiyama–Tanigawa’s conjecture, assuming the generalized Riemann hypothesis for all the symmetric power L-functions attached to f. In the third part, we give an application of discrepancy to a proof of joint universality of L-functions.