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

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.