Mean values and variances of the digits of $1/p$

Let $p\ge 3$ 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 $l$ of the period is the (multiplicative) order of $b$ mod $p$.

In the cases $l=p-1$ and $l=(p-1)/2$, formulas for the variance of the digits of a period were given previously.

These formulas involved Dedekind sums, class numbers of imaginary quadratic number fields, and generalized Bernoulli numbers.

In the present paper we develop a theory of this kind for $l=(p-1)/2^m$, $m\ge 1$, which covers the special case $l=(p-1)/2$.