The sum of the real parts of the zeros of the entire functions {1 + 2^z + ... + n^z: n>=2}

Abstract

We have proved that the sum of the real parts of the zeros of each partial sum 1+2^z+...+n^z of the Riemann zeta function is bounded for all integer n>= 2. If we take into account that the numerical experiences say us that, except for n = 2, their zeros are not located symmetrically with respect to the imaginary axis, this property may be considered as a surprising fact.