Fourier–Dedekind sums and an extension of Rademacher reciprocity

Tsukerman, Emmanuel

doi:10.1007/s11139-014-9555-x

Cited by 7 publications

(7 citation statements)

References 6 publications

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“…is the generalized Fourier-Dedekind sum (Definition 6.0.1). This motivates us to prove our next main result which is a generalization of existing results (see [8,14,25,6,5,22]).…”

Section: Resultssupporting

confidence: 60%

An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

Kiran¹

2021

Preprint

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In 1857 Sylvester established an elegant theory that certain counting functions, which he termed denumerants, are quasipolynomials by decomposing them into a polynomial part and a finite set of periodic parts. Each component of the decomposition, called a Sylvester wave, corresponds to a root of unity (the polynomial part corresponding to 1). Recently several researchers, using either combinatorial or complex analytic techniques, obtained explicit formulas for the waves. In this work, we develop an algebraic approach to the Sylvester's theory. Our methodology essentially relies on deriving q-partial fractions of the generating functions of the denumerants, and thereby obtain new and explicit formulas for the waves. The formulas we obtain are expressed in terms of reciprocal polynomial of the degenerate Bernoulli numbers and inverse DFT of a generalized Fourier-Dedekind sum. Further, we also prove certain reciprocity theorems of the generalized Fourier-Dedekind sums and a structure result on the top-order terms of the waves. The proofs rely on our symbolic evaluation operator and our far reaching generalization of the Heaviside's cover-up method for partial fractions.

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“…is the generalized Fourier-Dedekind sum (Definition 6.0.1). This motivates us to prove our next main result which is a generalization of existing results (see [8,14,25,6,5,22]).…”

Section: Resultssupporting

confidence: 60%

An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

Kiran¹

2021

Preprint

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“…. , a n is coprime to a 1 ; see also [2,23]. The expressions below for γ m (a) with m > 1 involve nontrivial "partial" Fourier-Dedekind sums that bear a resemblance to Ramanujan's sum, i.e.…”

Section: The Coefficients Of the Laurent Seriesmentioning

confidence: 99%

The Hilbert series and a-invariant of circle invariants

Cowie

Herbig

Herden

et al. 2019

Journal of Pure and Applied Algebra

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Let V be a finite-dimensional representation of the complex circle C × determined by a weight vector a ∈ Z n . We study the Hilbert series Hilba(t) of the graded algebra C[V ] C × a of polynomial C × -invariants in terms of the weight vector a of the C × -action. In particular, we give explicit formulas for Hilba(t) as well as the first four coefficients of the Laurent expansion of Hilba(t) at t = 1. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomial that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of C[V ] C × a in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.

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“…Here we prove a bound on the second smallest and second largest values, which improves on bounds given in Section 6 of [5].…”

Section: Smallest Values Of Inversion Numbermentioning

confidence: 70%

Dedekind sums s(a, b) and inversions modulo b

Chen

Dunn

Hewett

et al. 2015

Int. J. Number Theory

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We introduce the inversion polynomial for Dedekind sums f b (x) = x 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 reciprocity formula.

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Fourier–Dedekind sums and an extension of Rademacher reciprocity

Cited by 7 publications

References 6 publications

An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

The Hilbert series and a-invariant of circle invariants

Dedekind sums s(a, b) and inversions modulo b

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