The Critical Strips of the Sums 1 + 2 z + ⋯ + n z

Abstract

We give a partition of the critical strip, associated with each partial sum $1+2^z+\cdots +n^z$ of the Riemann zeta function for Re $z<-1$ formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.