Accumulation points of the sets of real parts of zeros of the partial sums of the Riemann zeta function

TL;DR

This paper investigates the accumulation points of the real parts of zeros of partial sums of the Riemann zeta function, revealing that zero is a common accumulation point and that the supremum of real parts is positive for all n>2.

Contribution

It establishes the existence of intervals of accumulation points for the zeros of partial sums of the Riemann zeta function and proves that the supremum of their real parts is positive.

Findings

Zero is an accumulation point for all sets of zeros.

The supremum of real parts of zeros is positive for all n>2.

Intervals of accumulation points include [-δₙ, bⁿ].

Abstract

It is shown that, for every integer n>2, there exists \delta_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function \zeta_{n} contains to the interval [-\delta_{n},b^{n}]. b^{n} is the supremum of the real parts of the zeros of \zeta_{n}. It is also demonstrated that b^{n} is positive for all n>2. It is also shown that 0 is an accumulation point common to all the sets P_{\zeta_{n}} wich are the sets of the real parts of the zeros of \zeta_{n}.