Digit frequencies and class numbers

Abstract

Let $p>3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $m$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $m=(p-1)/2$, the frequency of each digit $k\in\{0,1,\ldots, b-1\}$ can be expressed in terms of generalized (first order) Bernoulli numbers. In some cases only Bernoulli numbers belonging to quadratic characters occur. This means that the frequencies can be written in terms of class numbers of imaginary quadratic number fields (the so-called class number case). In the present paper we classify the numbers $p$ and $b$ falling under the class number case. We also highlight one of the simplest examples not falling under this case ($p\equiv 1$ mod $4$, $b=10$) and the most complex example of the class number case. Moreover, we show that knowing the frequency of each $k$ is equivalent to knowing the respective Bernoulli numbers.