Dedekind sums take each value infinitely many times

TL;DR

This paper proves that each value of the Dedekind sum is attained infinitely often by constructing sequences of pairs with increasing denominator, demonstrating the sum's value distribution is dense in a certain sense.

Contribution

It establishes that every Dedekind sum value occurs infinitely many times with unbounded denominators, revealing new insights into the sum's value distribution.

Findings

Dedekind sums take each value infinitely many times

Sequences of pairs with increasing denominators can produce the same sum value

The distribution of Dedekind sum values is dense in a certain sense

Abstract

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 $i\in \Bbb N$.

Equations

$$s(a,b)=\sum _{k=1}^b((k/b))((ak/b)).$$

$$((x))=\{\begin{array}{ll}x-\lfloor x\rfloor -1/2& \text{mbox}ifx\in \mathbb{R}\setminus \mathbb{Z};\\ 0& \text{mbox}ifx\in \mathbb{Z}\end{array}$$

$$S(a,b)=12s(a,b)$$

$$S(a+jb,b)=r\text{mbox}forallj\in \mathbb{Z}.$$

$$x=[0,c_1,c_2,\dots ],$$

$$x=[0,c_1,\dots ,c_n]$$

$$x=[0,\overline{c_1,\dots ,c_L}]=[0,c_1,\dots ,c_L,c_1,\dots ,c_L,\dots ].$$

$$S(p_k,q_k)=S(p_{L-1},q_{L-1}).$$

$$S(p_k,q_k)=\sum _{j=1}^L(-1{)}^{j-1}{c}_j+x+x^{\prime },$$

$$a/b=[0,c_1,\dots ,c_n]$$

$$x=[0,\overline{c_1,\dots ,c_n,c}].$$

$$S(p_k,q_k)=S(p_{L-1},q_{L-1})=S(p_n,q_n)=S(a,b)$$

$$x=[0,\overline{c_1,\dots ,c_{n-1},c_n-1,1,1}].$$

$$S(p_k,q_k)=S(p_{L-1},q_{L-1})=S(p_{n+1},q_{n+1})=S(a,b)$$

Full text

1. Introduction and result

Let $a$ be an integer, $b$ a natural number, and $(a,b)=1$. The classical Dedekind sum $s(a,b)$ is defined by

[TABLE]

Here

[TABLE]

(see [10, p. 1]).

Originally, Dedekind sums appeared in the theory of modular forms (see [2]). But these sums have also interesting applications in in a number of other fields, so in connection with class numbers, lattice point problems, topology, and algebraic geometry (see [3, 8, 10, 11]). Starting with Rademacher [9], several authors have studied the distribution of Dedekind sums (for instance, [4, 6, 12]).

It is often more convenient to work with

[TABLE]

instead. We call $S(a,b)$ a normalized Dedekind sum.

Let $r$ be a rational number such that there exist $a\in\mathbb{Z}$ and $b\in\mathbb{N}$, $(a,b)=1$ with $S(a,b)=r$. Then

[TABLE]

Accordingly, the value $r$ is taken infinitely many times in a trivial sense ($b$ fixed, $a$ running through a congruence class mod $b$).

In the present paper, however, we say that the value $r$ is taken infinitely many times if, and only if, there exists a sequence $(a_{i},b_{i})$, $i\in\mathbb{N}$, such that $b_{i}\to\infty$ as $i\to\infty$ and $S(a_{i},b_{i})=r$ for all $i\in\mathbb{N}$.

The only possible $r\in\mathbb{Z}$ that can be the value of a normalized Dedekind sum is $r=0$. The value [math] is taken infinitely many times since $S(a,a^{2}+1)=0$ for all $a\in\mathbb{N}$. This is well-known (see [10, p. 28]). Our main result is the following theorem.

Theorem 1

Let $r\in\mathbb{Q}$ be the value of a normalized Dedekind sum. Then the value $r$ is taken infinitely many times.

2. The proof

Let $x$ be a real number, $0<x<1$. We consider the regular continued fraction expansion

[TABLE]

where the $c_{i}$ are natural numbers. This expansion is finite, if, and only if, $x\in\mathbb{Q}$. In this case it has the form

[TABLE]

with $n\geq 1$ and $c_{n}\geq 2$. In the present setting the only irrational numbers of interest are quadratic irrationals. A number $x$ is a quadratic irrational if, and only if, its continued fraction expansion is infinite and periodic. We need only quadratic irrationals that are nearly purely periodic, i.e.,

[TABLE]

Let $p_{k}/q_{k}$, $k\geq 0$, be the $k$th convergent of $x$. The convergents are defined recursively in a well-known way (see [7, p. 250]). In particular, $p_{k}\in\mathbb{Z}$, $q_{k}\in\mathbb{N}$, and $(p_{k},q_{k})=1$ for all $k\geq 0$. Hence $S(p_{k},q_{k})$ is the value of a normalized Dedekind sum. If $x=a/b=[0,c_{1},\ldots,c_{n}]$ is rational, $a,b\in\mathbb{N}$, $(a,b)=1$, then $p_{n}=a$, $q_{n}=b$ (and so $p_{n}/q_{n}=x$). Otherwise, $q_{k}$ tends to infinity for $k\to\infty$.

The core of the proof of Theorem 1 is the following lemma, which is one of the main results of [5].

Lemma 1

Let $x=[0,\overline{c_{1},\ldots,c_{L}}]$ be a quadratic irrational with odd period length $L$. If $k\geq 0$, $k\equiv L-1\mod 2L$, then

[TABLE]

Remark. The constant value $S(p_{k},q_{k})$, $k\equiv L-1\mod 2L$, takes the form

[TABLE]

where $x^{\prime}$ is the algebraic conjugate of $x$ (see [5]).

Proof of Theorem 1. Let $S(a,b)$ be the value of a normalized Dedekind sum. By (1), we may assume that $0\leq a<b$, so $0\leq a/b<1$. Due to the remark that precedes Theorem 1, we suppose that $S(a,b)\neq 0$. Then $a/b\neq 0$. Let

[TABLE]

be the continued fraction expansion of $a/b$. We have $n\geq 1$ and, in particular, $c_{n}\geq 2$.

Case 1: $n$ is even. Choose an arbitrary natural number $c$ and define

[TABLE]

So this quadratic irrational has the odd number $L=n+1$ as a period length. By Lemma 1, the convergents of $x$ satisfy

[TABLE]

for $k\geq 0$, $k\equiv L-1\mod 2L$. Since $q_{k}\to\infty$ for $k\to\infty$, the value $S(a,b)$ is taken infinitely many times.

Case 2: $n$ is odd. We write $a/b=[0,c_{1},\ldots,c_{n-1},c_{n}-1,1]$ and put

[TABLE]

So the odd number $L=n+2$ is a period length of $x$. Observe that $p_{n+1}=a,q_{n+1}=b$. We obtain

[TABLE]

for $k\geq 0$, $k\equiv L-1\mod 2L$. This gives the same result as in Case 1. $\Box$

Example. Let $a=5$, $b=14$. Then $S(a,b)=18/7$ and $a/b=[0,2,1,4]$. Here $n=3$, so Case 2 of the proof applies. Accordingly, we define $x=[0,\overline{2,1,3,1,1}]=-5/7+\sqrt{226}/14$ and have $L=5$. We obtain $p_{4}=a,q_{4}=b$ and, for instance, $p_{14}=4535,q_{14}=12614$, $p_{24}=4090565$, $q_{24}=11377814$, $p_{34}=3689685095$, $q_{34}=10262775614$, where $14$, $24$ and $34$ are $\equiv L-1\mod 2L$. Indeed, $S(p_{34},q_{34})=S(p_{24},q_{24})=S(p_{14},q_{14})=S(a,b)=18/7$. From (2) we also obtain $S(a,b)=2-1+3-1+1-10/7=18/7$.

Remarks.

1. The proof of Theorem 1 has exhibited a sequence $(a_{i},b_{i})$ such that $b_{i}\to\infty$ and $S(a_{i},b_{i})=S(a,b)$ for all $i\in\mathbb{N}$. The sequence $b_{i}$, however, grows exponentially in $i$. This is a consequence of the exponential growth of the denominators $q_{k}$ of the convergents of $x$. Accordingly, the set of the numbers $b_{i}$ is rather thin within the set $\mathbb{N}$. In a number of special cases the author could establish a sequence $(a_{i},b_{i})$ of this kind such that $b_{i}$ is a polynomial of degree 4 in $i$ — and so the set of the numbers $b_{i}$ is considerably denser in $\mathbb{N}$.

2. It would be interesting to know more about the density of the set of all suitable numbers $b_{i}$. In our case, the following values of $(a_{i},b_{i})$ with $b_{i}<1000$ yield $S(a_{i},b_{i})=18/7$: $(5,14),(27,70),(13,119),(31,259),(157,406),(47,455),(293,707),(111,854)$. This suggests that the set of all suitable $b_{i}$ could be relatively dense in $\mathbb{N}$.