Stern’s type congruences for $L(-k,\chi )$

Abstract

We prove several Stern’s type congruences for Dirichlet $L$-function $L(-k,\chi )$. For example, suppose that $p$ is an odd prime, $m\ge 2$ and $\chi$ is a character with the conductor $p^m$. If $1-\chi (a)a^{k+1}$ is not prime to $p$ for each $1\le a\le p-1$, then we always have

$$\begin{aligned} |\mathcal L_{k+(p-1)h,\chi }-\mathcal L_{k,\chi }|_p=\frac{1}{p}|h|_p, \end{aligned}$$

where $|\cdot |_p$ denotes the $p$-adic norm and

$$\begin{aligned} \mathcal L_{k,\chi }=(1-\chi (p+1))L(-k,\chi ). \end{aligned}$$