On Li's criterion for the Riemann hypothesis for the Selberg class

Text. In this paper, we shall prove a generalization of Li's positivity criterion for the Riemann hypothesis for the extended Selberg class with an Euler sum. We shall also obtain two arithmetic expressions for Li's constants lambda(F)(n) = Sigma(rho)* (1 - 1/p)(n)), where the sum is taken over all non-trivial zeros of the function F and the * indicates that the sum is taken in the sense of the limit as T -> infinity of the sum over rho with |im rho| <= T. The first expression of lambda(F)(n), for functions in the extended Selberg class, having an Euler sum is given terms of analogues of Stieltjes constants (up to some gamma factors). The second expression, for functions in the Selberg class, non-vanishing on the line Re s = 1, is given in terms of a certain limit of the sum over primes. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EwDtXrkuwxA. (C) 2009 Elsevier Inc. All rights reserved.