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

Kurt Girstmair

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.

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

Abstract

Let be a prime and an integer such that does not divide . Then has a periodic digit expansion with respect to the basis . The length of the period is the (multiplicative) order of mod . In the case 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 . If mod 4 a Dedekind sum and the class number of occur in the respective formula. If 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.
Paper Structure (2 sections, 4 theorems, 57 equations)

This paper contains 2 sections, 4 theorems, 57 equations.

Table of Contents

  1. 1. Introduction and results
  2. 2. Proofs

Key Result

Theorem 1

Let $p\equiv 3$ mod $4$, $p>3$, and suppose that $b$ has the order $q=(p-1)/2$ mod $p$. Then

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2