Sur l'infimum des parties r\'eelles des z\'eros des sommes partielles de la fonction z\^eta de Riemann

TL;DR

This paper investigates the asymptotic behavior of the infimum of the real parts of zeros of partial sums of Riemann's zeta function, revealing a relationship involving the number of terms and the natural logarithm of 2.

Contribution

It establishes the asymptotic equivalence of the infimum of real parts of zeros with a formula involving the number of terms and logarithm of 2, providing new insight into partial sums of the zeta function.

Findings

The infimum of real parts of zeros approaches the negative of the number of terms times log(2).

As the number of terms increases, the lower bound of zeros' real parts behaves predictably.

The result links zero distribution of partial sums to classical constants and growth rates.

Abstract

The greatest lower bound of the real parts of the roots of a partial sum of the Dirichlet series of Riemann's zeta function is asymptotically equivalent to the opposite of the number of terms of this sum, multiplied by the Napierian logarithm of 2, when this number of terms tends to infinity.