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 ) ≡ 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 ( 2 z ) + ⋯ + f ( n z ) = 0 . We determine the density of the zeros of G n ( 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 G n ( z ) inside certain rectangles in the critical strip.