A note on large values of Dirichlet $L$-functions for characters of fixed order at $1/2<\sigma \le 1$

Abstract

In this note, we use a simple argument to show the existence of large values of conjecturally sharp size for Dirichlet $L$-functions attached to primitive characters of fixed order at $\sigma \in (1/2,1]$. More precisely, for every fixed integer $g\ge 2$ we prove the existence of a primitive character $\chi$ of order $g$ and conductor $Q\asymp x$ such that $|L(1,\chi )|\ge e^{\gamma }\left(\log \log x+\log \log \log x-\log (2\log g)+o(1)\right).$ We also show that for every fixed $1/2<\sigma <1$ there exists a primitive character $\chi$ of order $g$ and conductor $Q\asymp x$ such that $\log |L(\sigma ,\chi )|\ge \left(C_g(\sigma )+o(1)\right)(\log x{)}^{1-\sigma }(\log \log x{)}^{-\sigma },$ for some explicit positive constant $C_g(\sigma ).$ Previously, such bounds were known only conditionally on the Generalized Riemann Hypothesis, and even then only in the special cases $g=2$ and $g=3$.

7 pages