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

Kurt Girstmair

arxiv: 2606.29930 · v1 · pith:B5H7ICHWnew · submitted 2026-06-29 · 🧮 math.NT

Mean values and variances of the digits of 1/p

Kurt Girstmair This is my paper

Pith reviewed 2026-06-30 05:29 UTC · model grok-4.3

classification 🧮 math.NT

keywords mean valuesvariancesdigits of 1/pperiodic expansionsDedekind sumsclass numbersgeneralized Bernoulli numbersmultiplicative order

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The pith

The variance of digits in the base-b expansion of 1/p admits closed-form expressions in Dedekind sums, class numbers, and generalized Bernoulli numbers when the period length equals (p-1)/2^m for m at least 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends earlier formulas for the mean and variance of digits in the repeating expansion of 1/p. It covers the case where the multiplicative order of b modulo p is exactly (p-1) divided by a power of 2. The resulting expressions rely on the same arithmetic objects that appeared in the full-period and half-period cases. A reader would care because the formulas turn long digit sums into exact evaluations that depend only on invariants of p.

Core claim

For a prime p not dividing b, when the period length l equals (p-1)/2^m with m at least 1, both the mean value and the variance of the digits in one full period of the base-b expansion of 1/p are given by explicit formulas involving Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers.

What carries the argument

Closed-form expressions for the variance of the digit sum over a full period, obtained by reducing the sum via the order l = (p-1)/2^m to combinations of Dedekind sums and class-number terms.

If this is right

The earlier results for full period l = p-1 and for l = (p-1)/2 become immediate special cases of the new theory.
Exact variances become computable for any prime whose order is (p-1) divided by a power of two, without enumerating the digits.
The same arithmetic invariants control both the mean and the variance across this family of periods.
The approach supplies a uniform method that works uniformly for all m at least 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to other proper divisors of p-1 that are not powers of two.
Closed forms of this type could be used to test conjectures on the statistical uniformity of digits for primes with restricted orders.
Links between digit variances and class numbers might produce new relations between quadratic fields and base-b expansions.

Load-bearing premise

The variance of the digits admits closed-form expressions in terms of Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers when the multiplicative order l equals (p-1)/2^m.

What would settle it

Pick a prime p such as 41 where (p-1)/4 = 10, fix base b=10, compute the exact variance of the ten digits in one period of 1/41 by direct addition, and compare the numerical value against the closed-form prediction from the Dedekind-sum expression.

read the original 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 $p$. 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 $l=(p-1)/2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the known variance formulas for digits in 1/p expansions from the full and half-period cases to the full family of periods (p-1)/2^m.

read the letter

The main takeaway is that Girstmair has carried the earlier variance calculations for the digits of 1/p in base b over to the periods l = (p-1)/2^m for all m ≥ 1. The abstract makes clear that the same objects—Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers—still deliver closed forms.

The work does what it sets out to do: it treats the general m case as a single theory rather than a collection of special cases. The fact that it recovers the m=1 situation as a check is useful. The approach stays inside the existing number-theoretic toolkit, so the extension looks mechanical but not trivial.

The obvious soft spot is that only the abstract is in front of us. Without the actual derivations it is impossible to see how the sums and class-number terms behave when m increases, or whether extra correction terms appear that were absent for m=0,1. That leaves the soundness claim provisional.

The paper is aimed at the small group of people already tracking explicit formulas for digit statistics in rational base-b expansions. A reader who knows the two earlier papers will see immediately what has been added; outsiders will not find much to use.

The central claim is narrow but well-defined and appears free of circularity or post-hoc fitting. I would send it to referees rather than desk-reject it, on the grounds that the program is coherent and the new formulas are stated explicitly enough to be checked.

Referee Report

0 major / 2 minor

Summary. The paper extends prior results on the mean and variance of digits in the base-b expansion of 1/p (p prime, b not divisible by p) to the case where the multiplicative order l of b modulo p equals (p-1)/2^m for integers m ≥ 1. It derives closed-form expressions for these quantities in terms of Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers, thereby covering and generalizing the previously treated cases l = p-1 and l = (p-1)/2.

Significance. If the derivations are correct, the work supplies a uniform number-theoretic framework for digit statistics over an infinite family of periods indexed by powers of 2. This strengthens the connection between periodic digit expansions and classical objects (Dedekind sums, class numbers, Bernoulli numbers) and may enable explicit computations or further arithmetic applications for these special periods.

minor comments (2)

[Abstract] The abstract states that formulas 'were given previously' for l = p-1 and l = (p-1)/2 but does not cite the specific references; adding these citations would improve context.
[Introduction] Notation for the generalized Bernoulli numbers and the precise range of m should be introduced explicitly in the introduction to avoid any ambiguity for readers unfamiliar with the prior literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. We are pleased that the work is viewed as providing a uniform framework connecting digit statistics to classical arithmetic objects.

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent number-theoretic objects

full rationale

The abstract states that closed-form expressions for mean values and variances are developed in terms of Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers for the period length l=(p-1)/2^m. These are standard, externally defined objects in number theory. The paper extends prior formulas for l=p-1 and l=(p-1)/2 without any indication that the target quantities are fitted parameters, self-defined, or reduced by construction to the inputs. No load-bearing self-citation chain or ansatz smuggling is exhibited in the given text. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information is given on free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.1-grok · 5663 in / 1211 out tokens · 36812 ms · 2026-06-30T05:29:57.420270+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references

[1]

M., Introduction to Analytic Number Theory

Apostol, T. M., Introduction to Analytic Number Theory. New York, 1976

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[2]

Chakraborty, K., Krishnamoorthy, K., On some symmetries of the basenexpansion of 1/m: the class number connection, Pacific J. Math. 319, 39–53 (2022)

2022
[3]

D., Kulisch, U., Liedl, R

Girstmair, K., Periodische Dezimalbr¨ uche – was nicht jeder dar¨ uber weiß, in: Beu- telspacher, A., Chatterji, S. D., Kulisch, U., Liedl, R. (eds.) Jahrbuch ¨Uberblicke Mathematik 1995, 163–179, Braunschweig, 1995. 9

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[4]

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Girstmair, K., Digit variance and Dedekind sums, J. Number Th. 65, 197–205 (1997)

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[5]

Girstmair, K., On the variance of the digits of 1/p, Ramanujan J. 70, Art. 27 (2026)

2026
[6]

Hirabayashi, M., Generalizations of Girstmair’s formula, Abh. Math. Sem. Univ. Hamburg 75, 83–95 (2005)

2005
[7]

208, 215–233 (2023)

Mizuno, Y., A certain character twisted average value of the digits of rational num- bers and the class numbers of imaginary quadratic fields, Acta Arith. 208, 215–233 (2023)

2023
[8]

Moree, P., Near-primitive roots, Funct. Approx. Comment. Math. 48, 133–145 (2013)

2013
[9]

R., Thangadurai, R., The class number ofQ( √−p) and digits of 1/p, Proc

Murty, M. R., Thangadurai, R., The class number ofQ( √−p) and digits of 1/p, Proc. Amer. Math. Soc. 139, 1277–1289 (2011)

2011
[10]

Pujahari, S., Saikia, N.,l-adic digits and class numbers of imaginary quadratic fields, Internat. J. Math. 35, Paper No. 2450041, 16 pp. (2024)

2024
[11]

Mathematical Association of America, 1972

Rademacher, H., Grosswald, E., Dedekind Sums. Mathematical Association of America, 1972

1972
[12]

71, 273–278 (1995)

Szmidt, J., Urbanowicz, J., Zagier, D., Congruences among generalized Bernoulli numbers, Acta Arith. 71, 273–278 (1995). Institut f¨ ur Mathematik Universit¨ at Innsbruck Technikerstr. 13/7 A-6020 Innsbruck, Austria Kurt.Girstmair@uibk.ac.at 10

1995

[1] [1]

M., Introduction to Analytic Number Theory

Apostol, T. M., Introduction to Analytic Number Theory. New York, 1976

1976

[2] [2]

Chakraborty, K., Krishnamoorthy, K., On some symmetries of the basenexpansion of 1/m: the class number connection, Pacific J. Math. 319, 39–53 (2022)

2022

[3] [3]

D., Kulisch, U., Liedl, R

Girstmair, K., Periodische Dezimalbr¨ uche – was nicht jeder dar¨ uber weiß, in: Beu- telspacher, A., Chatterji, S. D., Kulisch, U., Liedl, R. (eds.) Jahrbuch ¨Uberblicke Mathematik 1995, 163–179, Braunschweig, 1995. 9

1995

[4] [4]

Number Th

Girstmair, K., Digit variance and Dedekind sums, J. Number Th. 65, 197–205 (1997)

1997

[5] [5]

Girstmair, K., On the variance of the digits of 1/p, Ramanujan J. 70, Art. 27 (2026)

2026

[6] [6]

Hirabayashi, M., Generalizations of Girstmair’s formula, Abh. Math. Sem. Univ. Hamburg 75, 83–95 (2005)

2005

[7] [7]

208, 215–233 (2023)

Mizuno, Y., A certain character twisted average value of the digits of rational num- bers and the class numbers of imaginary quadratic fields, Acta Arith. 208, 215–233 (2023)

2023

[8] [8]

Moree, P., Near-primitive roots, Funct. Approx. Comment. Math. 48, 133–145 (2013)

2013

[9] [9]

R., Thangadurai, R., The class number ofQ( √−p) and digits of 1/p, Proc

Murty, M. R., Thangadurai, R., The class number ofQ( √−p) and digits of 1/p, Proc. Amer. Math. Soc. 139, 1277–1289 (2011)

2011

[10] [10]

Pujahari, S., Saikia, N.,l-adic digits and class numbers of imaginary quadratic fields, Internat. J. Math. 35, Paper No. 2450041, 16 pp. (2024)

2024

[11] [11]

Mathematical Association of America, 1972

Rademacher, H., Grosswald, E., Dedekind Sums. Mathematical Association of America, 1972

1972

[12] [12]

71, 273–278 (1995)

Szmidt, J., Urbanowicz, J., Zagier, D., Congruences among generalized Bernoulli numbers, Acta Arith. 71, 273–278 (1995). Institut f¨ ur Mathematik Universit¨ at Innsbruck Technikerstr. 13/7 A-6020 Innsbruck, Austria Kurt.Girstmair@uibk.ac.at 10

1995