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

AbstractIn this paper we study the distribution of zeros of each entire function of the sequence {Gn(z)≡1+2z+⋯+nz:n⩾2}, which approaches the Riemann zeta function for Rez<−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 Gn(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