On Dedekind sums with equal values (Q2789376)

The paper under review considers the classical Dedekind sum defined for \(n\in\mathbb N\), \(m\in\mathbb Z\) with \((m,n)=1\) by NEWLINE\[NEWLINEs(m,n)=\sum_{k=1}^n\left(\left(\frac{k}{n}\right)\right)\left(\left(\frac{mk}{n}\right)\right),NEWLINE\]NEWLINE where \(((\cdots))\) is the function defined by NEWLINE\[NEWLINE(t)=\begin{cases} t-[t]-\frac{1}{2}&\quad \text{if }t\in\mathbb R\setminus\mathbb Z,\\ 0 &\quad\text{if }t\in\mathbb Z,\end{cases}NEWLINE\]NEWLINE and asks when these sums for different \(m\)'s are equal. Two cases of \(n\) are treated in this paper, namely, \(n\) a prime power and \(n\) square-free.NEWLINENEWLINEFor the prime power case, we have the following theorem, which is a weaker version of Rademacher's result.NEWLINENEWLINETheorem 1. Let \(d,n\in\mathbb N\), \(m\in\mathbb Z\) with \((m,n)=1\) and \(\varepsilon\in\{\pm 1\}\). Then NEWLINE\[NEWLINES(\varepsilon+dnm,dn^2)=\varepsilon\left(\frac{2}{dn^2}+d-3\right),NEWLINE\]NEWLINE where \(S(m,n)=12s(m,n)\). A different proof of Theorem 1 (Rademacher's) using the three-term relation for Dedekind sums is given, and three corollaries of this theorem are also stated.NEWLINENEWLINEFor the square-free case, the following holds.NEWLINENEWLINECorollary 4. Let \(t\in\mathbb N\) and \(t^2+4=qk^2\), where \(q\) is square-free and \(k\in\mathbb Z\). Let \(p_1,\dots,p_r\) be distinct prime numbers \(\geq 3\) such that \(p_j\nmid k\) and \((\frac{q}{p_j})\) for \(j=1,\dots,r\), where \((\frac{\cdot\cdot}{\cdot\cdot})\) is the Legendre symbol. Put \(n=p_1\cdots p_r\). Then there are \(2^r\) distinct numbers \(m\) with \(0\leq m<n\) and \((m,n)=1\) such that NEWLINE\[NEWLINES(1+mt,\,nt)=\frac{2}{nt}+\frac{t}{n}-3.NEWLINE\]NEWLINE The proof is given by using a theorem that was obtained by the author in [Int. J. Number Theory 10, No. 5, 1241--1244 (2014; Zbl 1296.11027)].