On the real projection of the zeros of 1+2^s+...+n^s

TL;DR

This paper investigates the accumulation points of the real projections of zeros of partial sums of the Riemann zeta function, suggesting an infinite density of zeros near certain lines in the complex plane.

Contribution

It provides new insights into the distribution and accumulation behavior of zeros of partial sums of the Riemann zeta function on the real axis.

Findings

Identification of potential accumulation points for zeros

Implication of infinitely many zeros near specific lines

Advancement in understanding zero distribution of partial sums

Abstract

In this paper, we focus on the existence of accumulation points of the subset defined by the real projection of the zeros of the partial sums of the Riemann zeta functions. That would imply the existence of an infinite amount of zeros of the partial sums of the Riemann zeta functions arbitrarily close to a line parallel to the imaginary axis passing through every accumulation point.