Extreme values of the Dirichlet $L$-functions at the critical points of the Riemann zeta function

TL;DR

This paper investigates the extreme values of Dirichlet L-functions at the critical points of the Riemann zeta function, revealing that their behavior aligns with values expected to the right of the critical line, regardless of the special nature of these points.

Contribution

It provides the first estimates of Dirichlet L-functions at the critical points of the Riemann zeta function, showing their extreme values mirror those to the right of the critical line.

Findings

Extreme values of |L(ρ',χ)| are estimated at zeta's critical points.

L-function behavior at these points is similar to behavior to the right of Re s=1.

Results support the hypothesis of behavior independence from the nature of critical points.

Abstract

We estimate large and small values of $|L(\rho',\chi)|$, where $\chi$ is a primitive character mod $q$ for $q>2$ and $\rho'$ runs over critical points of the Riemann zeta function in the right half of the one-line, that is, the points where $\zeta'(\rho')=0$ and $1\leq\Re \rho'$. It would be interesting to study how a certain Dirichlet $L$-function behaves at the critical points of the Riemann zeta function. We expect extreme values that an $L$-function would take at the critical points of the Riemann zeta function to be very close to the extreme values that the $L$-function would otherwise take to the right of the vertical line $\Re s=1 $. That is, an $L$-function is expected to behave in a manner that is independent of the nature of the points that are special with respect to the Riemann zeta function. The results obtained in this paper corroborate this behavior.