We consider generalized Dedekind sums in dimension $n$, for fixed $n$-tuple of natural numbers, defined as sum of products of values of periodic Bernoulli functions. This includes the higher dimensional Dedekind sums of Zagier and Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind sums. These are realized as coefficients of Todd series of lattice cones and satisfy reciprocity law from the cocycle property of Todd series. Using iterated residue formula, we compute the coefficient of the decomposition of of the Todd series corresponding to a nonsingular decomposition of the lattice cone defining the Dedekind sums. We associate a Laurent polynomial which is added to generalized Dedekind sums of fixed index to make their denominators bounded. We give explicitly the denominator in terms of Bernoulli numbers. This generalizes the role played by the rational function given by the difference of the Rademacher function and the classical Dedekind sums. We associate an exponential sum to the generalized Dedekind sums using the integrality of the generalized Rademacher function. We show that this exponential sum has a nontrivial bound that is sufficient to fulfill Weyl's equidistribution criterion and thus the fractional part of the generalized Dedekind sums are equidistributed. As an example, for a 3 dimensional case and Zagier's higher dimensional generalization of Dedekind sums, we compute the Laurent polynomials associated.
We give explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind-Rademacher sums, which are polynomial-time computable finite Fourier series. As a by-product we rederive a reciprocity law for these sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind-Rademach…
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 generators. A remarkable feature of Fourier-Dedekind sums is that they satisfy a reciprocity law called Rademacher reciprocity. In this paper, we study several aspects of Fourier-Dedekind sums: properties of general Fourier-Dedekind sums, extensions of the reciprocity law, average behavior of Fourier-Dedekind sums, and finally, extrema of 2-dimensional Fourier-Dede…
We express the number of lattice points inside certain simplices via Dedekind-Rademacher sums. As an application, we prove a conjecture of Kronheimer and Mrowka in the special case of Brieskorn spheres (with at most 4 singular fibers). This conjecture relates the Euler characteristic of the Seiberg-Witten-Floer homology to the Casson invariant.
The values of the so-called {\em Dedekind--Rademacher cocycle} at certain real quadratic arguments are shown to be global $p$-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by conjectures of Darmon, Dasgupta, and Vonk. The strategy for proving this result combines an approach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight one in the anti-parallel direction, building on the techniques in earlier work of Betina, Dimitrov, and Pozzi.
We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casson-Walker invariant of a lens space equals the average of its Heegaard-Floer correction terms. Additionally, we find an obstruction for the equality and equality with opposite sign, of two correction terms of the same lens space. Using this obstruction we are able to derive an opt…
We introduce and study the \emph{Rademacher-Carlitz polynomial} \[ \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1} u^{\fl{\frac{ka + t}{b}}} v^k \] where $a, b \in \Z_{>0}$, $s, t \in \R$, and $u$ and $v$ are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view $\RC(u, v, s, t, a, b)$ as a polynomial analogue (in the sense of Carlitz) of the \emph{Dedekind-Rademacher sum} \[ \r_t(a,b) := \sum_{k=0}^{b-1}\left(\left(\frac{ka+t}{b} \right)\right) \left(\left(\frac{k}{b} \right)\right), \] which ap…
We prove the simultaneous multiplication formulas for Apostol-Bernoulli polynomials and generalized Frobenius-Euler polynomials. These formulas contain Dedekind-Rademacher sums, Apostol-Dedekind sums and Fourier-Dedekind sums.
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, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers. This problem also goes by the name of the `coin exchange problem'. Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topolo…
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, and on the other hand it is given by a cotangent Laurent expansion.
Dedekind sums occur in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider poly-Dedekind sums which are obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and pr…
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 exponential sum using the cocycle property of the Todd series of 2d cones and the nonsingular cone decomposition along with the continued fraction of q/p. Its Weil bound is given for the modulus q, applying the purity theorem of the cohomology of the related l-adic sheaf due to Denef and Loeser. The Weil type bound of Denef and Loeser fulfills the Weyl's equidistribu…
We solve an open problem proposed in the book ``Computing the continuous discretely" written by Matthias Beck and Sinai Robins. That is, we proposed a polynomial time algorithm for calculating Fourier-Dedekind sums. The algorithm is simple modular Barvinok's simplicial cone decomposition. It can be easily adapted into De Leora et. al.'s LattE package, which gives a nice implimentation of Barvinok's polynomial time algorithm.
Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain …
We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics.
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
The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums, and give some exact computational formulae for them by using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function.
Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither "too big" nor "too small." We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation 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, Con…
We use the action of Atkin-Lehner operators to generate a family of reciprocity formulas for newform Dedekind sums. This family of reciprocity formulas provides symmetries which we use to investigate the kernel of these Dedekind sums.
The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind type DC sums, which are obtained from the Dedekind type DC sums by…
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.
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 in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of Kohnen, which gives a new proof of the density of values of classical Dedekind sums.
We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.
For a positive integer k and an arbitrary integer h, the Dedekind sum s(h,k) was first studied by Dedekind because of the prominent role it plays in the transformation theory of the Dedekind eta-function, which is a modular form of weight 1/2 for the full modular group SL_2(Z). There is an extensive literature about the Dedekind sums. Rademacher [8] has written an introductory book on the subject.
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 $p\ge 5$.
it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.
In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.
We obtain new trigonometric identities, which are some product-to-sum type formulas for the higher derivative of the cotangent and cosecant functions. Further, from specializations of our formulas, we derive not only various known reciprocity laws of generalized Dedekind sums but also new reciprocity laws of generalized Dedekind sums.
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 \[ \sum_{k \text{mod} a} \prod_{j=1}^d \cot^{(m_j)} π(\frac{k a_j}{a} + z_j ) . \] Here the sum is taken over all k mod a for which the summand is not singular. These Dedekind cotangent sums generalize and unify various arithmetic sums introduced by Dedekind, Rademacher, Apostol, Carlitz, Zagier, Berndt, Meyer, Sczech, and Dieter. Generalized Dedekind sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and …
As a consequence of their work, Bruce C. Berndt and Ronald J. Evans in 1977 and Larry Joel Goldstein and Michael Razar in 1976 obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. Here we focus only on this formula and give a new short and simple proof of it. We finally express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums.
We introduce the inversion polynomial for Dedekind sums $f_b(x)=\sum x^{\operatorname{inv}(a,b)}$ to study the number of $s(a,b)$ which have the same value for given $b$. We prove several properties of this polynomial and present some conjectures. We also introduce connections between Kloosterman sums and the inversion polynomial evaluated at particular roots of unity. Finally, we improve on previously known bounds for the second highest value of the Dedekind sum and provide a conjecture for a possible generalization. Lastly, we include a new restriction on equal Dedekind sums based on the rec…
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 $i\in \Bbb N$.
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 $\varepsilon$.
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 cas…
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 $s(a_1,b)$ and $s(a_2,b)$ given in [Jabuka, Robins and Wang, When are two Dedekind sums equal? Internat. J. Number Theory 7: (2011) 2197--2202] is equivalent to $12s(a_1,b)-12s(a_2,b) \in \mathbb{Z}$. In this note, we give a new proof of this result and then find two additional necessary and sufficient conditions for $12s(a_1,b)-12s(a_2,b) \in 2\mathbb{Z}, 4\mathb…
Let $p_1,p_2,\dots,p_n, a_1,a_2,\dots,a_n \in \N$, $x_1,x_2,\dots,x_n \in \R$, and denote the $k$th periodized Bernoulli polynomial by $\B_k(x)$. We study expressions of the form \[ \sum_{h \bmod{a_k}} \ \prod_{\substack{i=1\\ i\not=k}}^{n} \ \B_{p_i}\left(a_i \frac{h+x_k}{a_k}-x_i\right). \] These \highlight{Bernoulli--Dedekind sums} generalize and unify various arithmetic sums introduced by Dedekind, Apostol, Carlitz, Rademacher, Sczech, Hall--Wilson--Zagier, and others. Generalized Dedekind sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and co…
We study the number of lattice points in integer dilates of the rational polytope $P = (x_1,...,x_n) \in \R_{\geq 0}^n : \sum_{k=1}^n x_k a_k \leq 1$, where $a_1,...,a_n$ are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers $a_1,...,a_n$, find the largest value of t (the Frobenius number) such that $m_1 a_1 + ... + m_n a_n = t$ has no solution in positive integers $m_1,...,m_n$. This is equivalent to the problem of finding the largest dilate tP such that the facet $\sum_{k=1}^n x_k a_k = t$ contains no …
We study a generalization of the classical Dedekind sum that incorporates two Dirichlet characters and develop properties that generalize those of the classical Dedekind sum. By calculating the Fourier transform of this generalized Dedekind sum, we obtain an explicit formula for its second moment. Finally, we derive upper and lower bounds for the second moment with nearly identical orders of magnitude.
Higher-dimensional Dedekind sums are defined as a generalization of a recent 1-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the frequency distribution of marked lattice points leads to new formulae in certain special cases, and to new bounds for the classical Dedekind sums. Upper bounds for the generalized Dedekind sums are defined in terms of 1-dimensional moments. In the classical two-dimensional case, the ratio of these sums to their upper bounds are cosines of angles between certain vectors of n-dimensional cones, conjectured to have a lar…
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.
Using a generalization due to Lerch [M. Lerch, Sur un théorème de Zolotarev. Bull. Intern. de l'Acad. François Joseph 3 (1896), 34-37] of a classical lemma of Zolotarev, employed in Zolotarev's proof of the law of quadratic reciprocity, we determine necessary and sufficient conditions for the difference of two Dedekind sums to be in $8\mathbb{Z}$. These yield new necessary conditions for equality of two Dedekind sums. In addition, we resolve a conjecture of Girstmair [Girstmair, Congruences mod 4 for the alternating sum of the partial quotients, arXiv: 1501.00655].
We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by the second author. Hence we obtain a polynomial-time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witte…
In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.
