The class number of $\mathbb {Q}(\sqrt {-p})$ and digits of $1/p$

Abstract:

Let $p$ be a prime number such that $p\equiv 1\pmod {r}$ for some integer $r >1$. Let $g>1$ be an integer such that $g$ has order $r$ in $\left (\mathbb {Z}/p\mathbb {Z}\right )^*$. Let \[ \frac 1p = \sum _{k=1}^\infty \frac {x_k}{g^k}\] be the $g$-adic expansion. Our result implies that the “average” digit in the $g$-adic expansion of $1/p$ is $(g-1)/2$ with error term involving the generalized Bernoulli numbers $B_{1,\chi }$ (where $\chi$ is a character modulo $p$ of order $r$ with $\chi (-1) = -1)$. Also, we study, using Bernoulli polynomials and Dirichlet $L$-functions, the set equidistribution modulo $1$ of the elements of the subgroup $H_n$ of $\left (\mathbb {Z}/{n\mathbb Z}\right )^*$ as $n\to \infty$ whenever $|H_n|/\sqrt {n} \to \infty$.