The mean value of the digits of $1/p$

Abstract

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 $p$. If $l$ is even, then the mean value of the digits of a period is just $(b-1)l/2$. The case of an odd length $l$ is more interesting. If $l=(p-1)/2^m$ is odd, the mean value of the digits of a period was given previously. This mean value involves generalized Bernoulli numbers. However, it is not clear how this result can be generalized to an arbitrary odd length $l$. In the present note we settle this case.