Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> be a prime and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q=p-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q=(p-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\equiv 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> mod 4 a Dedekind sum and the class number of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb Q(\sqrt{-p})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mo>-</mo> <mi>p</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> occur in the respective formula. If <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p\equiv 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> 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

期刊论文
发表日期：

2026年5月19日

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Kurt Girstmair

摘要

Let $p>3$ $p > 3$ be a prime and $b\ge 2$ $b \geq 2$ 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 $q=p-1$ $q = p - 1$ 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 $q=(p-1)/2$ $q = (p - 1) / 2$ . If $p\equiv 3$ $p \equiv 3$ mod 4 a Dedekind sum and the class number of $\mathbb Q(\sqrt{-p})$ $Q (\sqrt{- p})$ occur in the respective formula. If $p\equiv 1$ $p \equiv 1$ 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.