On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

In this paper we study the distribution of zeros of each entire function of the sequence {G(n)(z) equivalent to 1 + 2(z) + ... + n(z): n >= 2}, which approaches the Riemann zeta function for Re z < -1, and is closely related to the solutions of the functional equations f(z) + f(2z) + ... + f(nz) = 0. We determine the density of the zeros of G,,(z) on the critical strip where they are Situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of Gn(z) inside certain rectangles in the critical strip. (C) 2008 Elsevier Inc. All rights reserved.