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Kurt Girstmair

Publications and source records attributed to Kurt Girstmair.

At least 37 records · Page 2Linked to original sources
arXiv2017

On the distribution of Dedekind sums

Dedekind sums have applications in quite a number of fields of mathematics. Therefore, their distribution has found considerable interest. This article gives a survey of several aspects of the distribution of these sums. In particular, it highlights results about the values of Dedekind sums, their density and uniform distribution. Further topics include mean values, large and small (absolute) values, and the behaviour of Dedekind sums near quadratic irrationals. The present paper can be considered as a supplement to the survey article [R. W. Bruggeman, On the distribution of Dedekind sums, Contemp. Math. 166 (1994), 197--210].

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arXiv2017

The rational part of a periodic continued fraction

Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The numbers $a$ and $\sqrt b$ are uniquely determined by $x$. In general it is difficult to say what the influence of a certain digit of the repeating block on the appearance of $x$ is. We highlight a noteworthy exception from this rule. Indeed, the magnitude of $2a$ is essentially determined by the last digit $c_n$ of the repeating block, the fractional part of $2a$, however, is independent of $c_n$. Of particular interest is the case $2a\in\mathbb Z$, which occurs if, and only if, the sequence $c_1,\ldots,c_{n-1}$ is symmetric.

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arXiv2017

Dedekind sums take each value infinitely many times

For $a\in \Bbb Z$ and $b\in\Bbb N$, $(a,b)=1$, let $s(a,b)$ denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs $(a_i,b_i)$, $i\in\Bbb N$, with $b_i$ tending to infinity as $i$ grows, such that $s(a_i,b_i)=s(a,b)$ for all $i\in \Bbb N$.

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arXiv2017

The largest values of Dedekind sums

Let $s(m,n)$ denote the classical \DED sum, where $n$ is a positive integer and $m\in\{0,1,\ldots, n-1\}$, $(m,n)=1$. For a given positive integer $k$, we describe a set of at most $k^2$ numbers $m$ for which $s(m,n)$ may be $\ge s(k,n)$, provided that $n$ is sufficiently large. For the numbers $m$ not in this set, $s(m,n)<s(k,n)$.

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arXiv2016

On the values of Dedekind sums

Let $S(a,b)=12s(a,b)$, where $s(a,b)$ denotes the classical Dedekind sum. For a given denominator $q\in \mathbb N$, we study the numerators $k\in\mathbb Z$ of the values $k/q$, $(k,q)=1$, of Dedekind sums $S(a,b)$. Our main result says that if $k$ is such a numerator, then the whole residue class of $k$ modulo $(q^2-1)q$ consists of numerators of this kind. This fact reduces the task of finding all possible numerators $k$ to that of finding representatives for finitely many residue classes modulo $(q^2-1)q$. By means of the proof of this result we have determined all possible numerators $k$ for $2\le q\le 50$, the case $q=1$ being trivial. The result of this search suggests a conjecture about all possible values $k/q$, $(k,q)=1$, of Dedekind sums $S(a,b)$ for an arbitrary $q\in\mathbb N$.

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arXiv2016

Equality of Dedekind sums modulo 24$\mathbb Z$

Let $S(a,b)=12s(a,b)$, where $s(a,b)$ denotes the classical Dedekind sum. In a recent note E. Tsukerman gave a necessary and sufficient condition for $S(a_1,b)-S(a_2,b)\in 8\mathbb Z$. In the present paper we show that this condition is equivalent to $S(a_1,b)-S(a_2,b)\in 24\mathbb Z$, provided that $9\nmid b$. Tsukerman also obtained a congruence mod 8 for $bT(a,b)$, where $T(a,b)$ is the alternating sum of the partial quotients of the continued fraction expansion of $a/b$. We show that the respective congruence holds mod $24$ if $3\nmid b$ and mod $72$ if $3\mid b$.

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arXiv2016

Dedekind sums in the $p$-adic number field

In a recent note W. Kohnen asks whether the values of Dedekind sums are dense in the field of $p$-adic numbers. The present paper answers this question. Dedekind sums do not approximate units of $\mathbb Z_2$ or $\mathbb Z_3$, so they are not dense in $\mathbb Q_2$ or $\mathbb Q_3$. But they are dense in $\mathbb Q_p$ if $p\ge 5$.

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arXiv2015

Moebius transforms and cyclic equations

We present a simple method for the construction of polynomials with cyclic Galois groups, hoping to encourage a reader with some background in algebra to make computations of his/her own.

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arXiv2015

Periodic continued fractions and Kronecker symbols

We study the Kronecker symbol $\left(\frac st\right)$ for the sequence of the convergents $s/t$ of a purely periodic continued fraction expansion. Whereas the corresponding sequence of Jacobi symbols is always periodic, it turns out that the sequence of Kronecker symbols may be aperiodic. Our main result describes the period length in the periodic case in terms of the period length of the sequence of Jacobi symbols and gives a necessary and sufficient condition for the occurrence of the aperiodic case.

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arXiv2015

Congruences mod 4 for the alternating sum of the partial quotients

We apply a technique used in $[$Tsukerman, Equality of Dedekind sums mod $\mathbb Z$, $2\mathbb Z$ and $4\mathbb Z$, arXiv:1408.3225] combined with the Barkan-Hickerson-Knuth-formula in order to obtain congruences mod $4$ for the alternating sum of the partial quotients of the continued fraction expansion of $a/b$, where $0<a<b$ are integers.

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arXiv2014

On Dedekind sums with equal values

Dedekind sums $s(m,n)$ occur in many fields of mathematics. Since $s(m_1,n)=s(m_2,n)$ if $m_1\equiv m_2$ mod $n$, it is natural to ask which of the Dedekind sums $s(m,n)$, $0\le m<n$, take equal values. So far no simple criterion is known by which the equality of $s(m_1,n)$ and $s(m_2,n)$ could be decided. In this note we show how to obtain non-obvious examples of equal Dedekind sums. We consider two cases which mark the extreme possibilities for the argument $n$, namely, $n$ a prime power and $n$ square-free. Whereas we can give a partial overview of equal Dedekind sums in the prime power case, such an overview seems to be much more difficult to obtain in the square-free case.

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arXiv2014

On the asymptotic behavior of Dedekind sums

Let $z$ be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums $S(p_k,q_k)$ belonging to convergents $p_k/q_k$ of the {\em regular} continued fraction expansion of $z$ with that of Dedekind sums $S(s_j/t_j)$ belonging to convergents $s_j/t_j$ of the {\em negative regular} continued fraction expansion of $z$. Whereas the three main cases of this behavior are closely related, a more detailed study of the most interesting case (in which the Dedekind sums remain bounded) exhibits some marked differences, since the cluster points depend on the respective periods of these expansions. We show in which cases cluster points of $S(s_j,t_j)$ can coincide with cluster points of $S(p_k,q_k)$. An important tool for our purpose is a criterion that says which convergents $s_j/t_j$ of $z$ are convergents $p_k/q_k$.

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arXiv2013

The number of Dedekind sums with equal fractional parts

In a previous it was shown that the Dedkind sums $12s(m,n)$ and $12s(x,n)$, $1\le m,x\le n$, $(m,n)=(x,n)=1$, are equal mod $\Z$ if, and only if, $(x-m)(xm-1)\equiv 0$ mod $n$. Here we determine the cardinality of numbers $x$ in the above range that satisfy this congruence for a given number $m$.

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arXiv2013

Approximation of rational numbers by Dedekind sums

Given a rational number $x$ and a bound $\varepsilon$, we exhibit $m,n$ such that $|x-12 s(m,n)|<\varepsilon$. Here $s(m,n)$ is the classical Dedekind sum and the parameters $m$ and $n$ are completely explicit in terms of $x$ and $\varepsilon$.

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arXiv2013

On a criterion for the equality of Dedekind Sums

In [3] it was shown that the Dedekond sums $s(m_1,n)$ and $s(m_2,n)$ are equal only if $(m_1m_2-1)(m_1-m_2)\equiv 0$ mod $n$. Here we show that the latter condition is equivalent to $12s(m_1,n)-12s(m_2,n)\in \Z$. In addition, we determine, for a given number $m_1$, the number of integers $m_2$ in the range $0\le m_2<n$, $(m_1,m_2)=1$, such that $12s(m_1,n)-12s(m_2,n)\in \Z$, provided that $n$ is square-free.

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arXiv2013

Dedekind sums with arguments near certain transcendental numbers

We study the asymptotic behaviour of the classical Dedekind sums $s(s_k/t_k)$ for the sequence of convergents $s_k/t_k$ $k\ge 0$, of the transcendental number \BD \sum_{j=0}^\infty\frac {1}{b^{2^j}},\ b\ge 3. \ED In particular, we show that there are infinitely many open intervals of constant length such that the sequence $s(s_k/t_k)$ has infinitely many transcendental cluster points in each interval.

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arXiv2013

The period length of Euler's number e

Let s_k/t_k, k>= 0, be the convergents of the continued fraction expansion of a real number x. We investigate the sequence of Jacobi symbols (s_k/t_k), k>= 0. We show that this sequence is purely periodic with shortest possible period length 24 for x=e=2.718281... and shortest possible period length 40 for x=e^2. Further, we make the first steps towards a general theory of such sequences of Jacobi symbols. For instance, we show that there are uncountably many numbers x such that this sequence has the period 1 (of length 1), and that every natural number L actually occurs as the shortest possible period length of some x.

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