Related papers: Equality of Dedekind sums mod $8 \mathbb{Z}$

In [Girstmair, A criterion for the equality of Dedekind sums mod $\mathbb{Z}$, Internat. J. Number Theory 10: (2014) 565--568], it was shown that the necessary condition $b \mid (a_1 a_2-1)(a_1-a_2)$ for equality of two dedekind sums…

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…

In this paper, for coprime numbers p and q we consider the well known Dedekind sums S(p,q) First, we give an improvement of the proof given by H. Rademacher and A. Whiteman, and we construct a new arithmetical proof for the reciprocity law

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,…

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…

Dedekind sums are arithmetic sums that were first introduced by Dedekind in the context of elliptic functions and modular forms, and later recognized to be surprisingly ubiquitous. Among the variations and generalizations introduced since,…

A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \ | \ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of…

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…

Dedekind sums are well-studied arithmetic sums, with values uniformly distributed on the unit interval. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of $SL(2,\mathbb{Z})$. We…

Let $S(a,b)$ denote the normalized Dedekind sum. We study the range of possible values for $S(a,b)=\frac{k}{q}$ with $\gcd(k,q)=1$. Girstmair proved local restrictions on $k$ depending on $q\pmod{12}$ and whether $q$ is a square and…

For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this…

Let a, a_1, ..., a_d be positive integers, m_1, ..., m_d nonnegative integers, and z_1, ..., z_d complex numbers. We study expressions of the form \[ \sum_{k \text{mod} a} \prod_{j=1}^d \cot^{(m_j)} \pi (\frac{k a_j}{a} + z_j ) . \] Here…

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes. In particular,…

In this paper, we study the generalized Dedekind-Rademacher sums considered by Hall, Wilson and Zagier. We establish a formula for the products of two Bernoulli functions. The proof relies on Parseval's formula, Hurwitz's formula, and…

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,…

Fourier-Dedekind sums are a generalization of Dedekind sums - important number-theoretical objects that arise in many areas of mathematics, including lattice point enumeration, signature defects of manifolds and pseudo random number…

We study reciprocity formulas for Dedekind sums associated with absolutely continuous functions, extending the classical Dedekind-Rademacher reciprocity formula. In particular, we treat the case of periodic Bernoulli functions. Our approach…

We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be same, by utilizing the character analogue of the Euler-MacLaurin summation formula.…

Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real $j$-invariant, the values of suitably normalized elliptic Dedekind sums are dense…

We obtain a new motivated proof of the reciprocity law for Dedekind sums by computing the constant coefficient of the Ehrhart polynomial for a rectangular triangle in two ways. On the one hand, the constant term is the Euler characteristic,…