arXiv subjects

Kurt Girstmair

Publications and source records attributed to Kurt Girstmair.

At least 19 recordsLinked to original sources

arXiv2026

The mean value 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 $p$. 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.

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arXiv2026

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 $l=(p-1)/2$.

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arXiv2026

Digit frequencies and class numbers

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 $m$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $m=(p-1)/2$, the frequency of each digit $k\in\{0,1,\ldots, b-1\}$ can be expressed in terms of generalized (first order) Bernoulli numbers. In some cases only Bernoulli numbers belonging to quadratic characters occur. This means that the frequencies can be written in terms of class numbers of imaginary quadratic number fields (the so-called class number case). In the present paper we classify the numbers $p$ and $b$ falling under the class number case. We also highlight one of the simplest examples not falling under this case ($p\equiv 1$ mod $4$, $b=10$) and the most complex example of the class number case. Moreover, we show that knowing the frequency of each $k$ is equivalent to knowing the respective Bernoulli numbers.

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arXiv2026

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

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.

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arXiv2026

On the decimal and octal digits of $1/p$

Let $p$ be a prime $\equiv 3$ mod 4, $p>3$, and suppose that 10 has the order $(p-1)/2$ mod p. Then $1/p$ has a decimal period of length $(p-1)/2$. We express the frequency of each digit $0,\ldots,9$ in this period in terms of the class numbers of two imaginary quadratic number fields. We also exhibit certain analogues of this result, so for the case that 10 is a primitive root mod $p$ and for the octal digits of $1/p$.

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arXiv2025

Power residues, digit expansions and relative class numbers

This is a survey of a connection between the distribution of certain power residues modulo $p$, $p$ a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields a number of results about the distribution of quadratic residues, for instance, the well-known fact that the interval $[0,p/2]$ contains more quadratic residues than nonresidues. This class number formula is also responsible for some properties of the digit expansions of numbers $m/p$, $p\NDIV m$. In a certain sense the results based on Dirichlet's formula can be extended to sixth power residues, where geometry plays an important role.

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arXiv2025

Cotangent power sums and character coordinates

We show that certain sums studied in two recent papers are basically character coordinates (as they are called in the literature). These sums involve values of Dirichlet characters and powers of $\cot(πk/n)$, $1\le k\le n-1$. We also show that a basic tool for the study of these sums was already given in 1987, in the form of the character coordinates of so-called cotangent numbers. By means of this tool, we obtain the results of the said papers in a simple and lucid way. We also show that the coefficients of the linear combinations used in the said papers are essentially the same.

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arXiv2025

Hecke's inverse cotangent numbers

In connection with Eisenstein series for the principal congruence subgroup $Γ(n)$, Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the $n$th cyclotomic field or its maximal real subfield. They arise from the well investigated {\em cotangent numbers} by matrix inversion, which is why we call them {\em inverse cotangent numbers}. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that $n$ is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums.

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arXiv2022

Units from square-roots of rational numbers

Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\, q^2$. We show that the units $>1$ of the ring $\mathbb Z[\sqrt{Dq^2/Q}]$ are connected with certain convergents of $\sqrt{D/Q}$. Among these units, the units of $\mathbb Z[\sqrt{DQ}]$ play a special role, inasmuch as they correspond to the convergents of $\sqrt{D/Q}$ that occur just before the end of each period. We also show that the last-mentioned units allow reading the (periodic) continued fraction expansion of certain quadratic irrationals from the (finite) continued fraction expansion of certain rational numbers.

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arXiv2022

On the equivalence of certain quadratic irrationals

This paper deals with quadratic irrationals of the form $m/q+\sqrt v$ for fixed positive integers $v$ and $q$, $v$ not a square, and varying integers $m$, $(m,q)=1$. Two numbers $m/q+\sqrt v$, $n/q+\sqrt v$ of this kind are equivalent (in a classical sense) if their continued fraction expansions can be written with the same period. We give a necessary and sufficient condition for the equivalence in terms of solutions of Pell's equation. Moreover, we determine the number of equivalence classes to which these quadratic irrationals belong.

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arXiv2021

On the equality of Dedekind sums

We show that deciding the equality of two Dedekind sums $S(c,b)$, $S(d,b)$ is equivalent to deciding whether a Dedekind sum defined by $b, c, d$ takes a certain value. By means of this result we construct infinite sequences of pairwise equal Dedekind sums. Moreover, we prove a result that says how many Dedekind sums $S(d,b)$, $1\le d\le b-1$, may be equal to a given $S(c,b)$ if $b$ is a square-free number.

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arXiv2020

Chebyshev polynomials and Galois groups of De Moivre polynomials

Let $n\ge 3$ be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree $n$ with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated only for prime numbers $n$ and under special assumptions. We describe the Galois groups for arbitrary odd $n\ge 3$ in the irreducible case, up to few exceptions. In addition, we express all zeros of such a polynomial as rational functions of three zeros, two of which are connected in a certain sense. These results are based on the reduction of the radical $$ \sqrt[n]{d+\sqrt R}, $$ whose degree is $2n$ in general, to irrationals of degree $\le n$. Such a reduction was given in a previous paper of the author. Here, however, we present a much simpler approach that is based on properties of Chebyshev polynomials.

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arXiv2020

Reducing radicals in the spirit of Euclid

Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's {\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whose degree is $2p$, as a polynomial function of irrationals of degrees $\le p$. In certain cases $y$ is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's algorithm.

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arXiv2019

The Petersson-Knopp identity and Farey neighbours

We study Dedekind sums $S(a,b)$ near Farey points of the interval $[0,b]$. Each of these Dedekind sums is connected with a set of other Dedekind sums by the Petersson-Knopp identity. In the case considered here, this identity has a very specific interpretation, inasmuch as each Dedekind occurring in this identity is close to a certain expected value. Conversely, each of these expected values occurs with a certain frequency, a frequency that is consistent with the Petersson-Knopp identity.

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arXiv2018

On a recent reciprocity formula for Dedekind sums

Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Recently, Du and Zhang proved the following reciprocity formula. If $a$ and $b$ are odd natural numbers, $(a,b)=1$, then $$ S(2a^*,b)+S(2b^*,a)=\frac{a^2+b^2+4}{2ab}-3, $$ where $aa^*\equiv 1\mod b$ and $bb^* \equiv 1 \mod a$. In this paper we show that this formula is a special case of a series of similar reciprocity formulas. Whereas Du and Zhang worked with the connection of Dedekind sums and values of $L$-series, our main tool is the three-term relation for Dedekind sums.

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arXiv2018

Inversion of two cyclotomic matrices

Let $n\ge 3$ be a square-free natural number. We explicitly describe the inverses of the matrices $$ (2\sin(2πjk^*/n))_{j,k} \enspace \mbox{ and }\enspace (2\cos(2πjk^*/n))_{j,k}, $$ where $k^*$ denotes a multiplicative inverse of $k$ mod $n$ and $j,k$ run through the set $\{l; 1\le l\le n/2, (l,n)=1\}$. These results are based on the theory of Gauss sums.

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arXiv2018

On the moduli of a Dedekind sum

Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Let $k/q$, $q\in \Bbb N$, $k\in \Bbb Z$, $(k,q)=1$, be the value of $S(a,b)$. In a previous paper we showed that there are pairs $(a_r,b_r)$, $r\in\Bbb N$, such that $S(a_r,b_r)=k/q$ for all $r\in \Bbb N$, the $b_r$'s growing in $r$ exponentially. Here we exhibit such a sequence with $b_r$ a polynomial of degree $4$ in $r$.

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arXiv2018

Linear relations among asymptotic frequencies in continued fractions

Let $H(m,d)$ denote the asymptotic frequency of the natural numbers $k\equiv d \mod m$ in the continued fraction expansions of almost all numbers $x\in[0,1)$. For a fixed number $m\ge 4$, we study $\mathbb Q$-linear relations among the numbers $H(m,d)$, $1\le d\le m-3$, i.e., vectors $(c_1,\ldots,c_{m-3})\in\mathbb Q^{m-3}$ such that $$ \sum_{d=1}^{m-3} c_dH(m,d)=0. $$ We restrict ourselves to the symmetric case $c_d=c_{m-2-d}$. In the end, we obtain a basis of the $\mathbb Q$-vector space of these relations for prime powers $m$ and for $m=pq$, where $p\ne q$ are primes.

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