On the variance of the digits of $1/p$

TL;DR

This paper investigates the variance of digits in the periodic expansion of 1/p in various bases, deriving formulas involving Dedekind sums and class numbers for specific cases of the period length.

Contribution

It extends previous work by deriving the variance formula for the case where the period length is half of p-1, involving advanced number theoretic functions.

Findings

Variance formula involves Dedekind sums for q=p-1

For q=(p-1)/2, formulas include class numbers and Bernoulli numbers

Results depend on the congruence of p modulo 4

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 $q$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $q=p-1$ a formula for the variance of the digits of a period was given previously. This formula involves a Dedekind sum. We determine the variance in the case $q=(p-1)/2$. If $p\equiv 3$ mod 4 a Dedekind sum and the class number of $\mathbb Q(\sqrt{-p})$ occur in the respective formula. If $p\equiv 1$ mod 4, the formula may be much more complex since it involves linear combinations of (possibly many) products of two Bernoulli numbers attached to odd characters.