Dedekind Sums Research Articles

A hybrid mean value of Dedekind sums and Kloosterman sums

In this paper, we use the properties of Gauss sums and analytic methods to study the hybrid mean value problems involving Dedekind sums and Kloosterman sums, and give an interesting asymptotic formula for it. At the same time, we can also use the similar methods to deal with the other related sums, such as Cochrane sums.

Dedekind sums with a parameter in finite fields

We first introduce the multiple Dedekind–Rademacher sum with a parameter in finite fields and establish its reciprocity law. We then construct an analog of the higher-dimensional Apostol–Dedekind sums, and establish their reciprocity laws using the parameterized Dedekind sum.

On the general Dedekind sums and two-term exponential sums.

We use the analytic methods and the properties of Gauss sums to study the computational problem of one kind hybrid mean value involving the general Dedekind sums and the two-term exponential sums, and give an interesting computational formula for it.

A hybrid mean value involving Dedekind sums and the general exponential sums.

The main purpose of this paper is using the analytic method, A. Weil's classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharp asymptotic formula for it.

The average signature of graph links

Abstract We compute the average Tristram–Levine signature of any graph link with positive weights in a three sphere. The main tools are Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.

On the mean value of the two-term Dedekind sums

The main purpose of this paper is, using the properties of Gauss sums, the estimate for character sums and the analytic method, to study the mean value of the two-term Dedekind sums and give an interesting asymptotic formula for it.

Some identities related to Dedekind sums and the second-order linear recurrence polynomials

In this paper, we use the elementary method and the reciprocity theorem of Dedekind sums to study the computational problem of one kind Dedekind sums, and give two interesting computational formulae related to Dedekind sums and the second-order linear recurrence polynomials.

An interesting identity and asymptotic formula related to the Dedekind sums

We use the analytic methods and the properties of Gauss sums to study one kind mean value problems involving the classical Dedekind sums, and give an interesting identity and asymptotic formula for it.

DENOMINATORS OF DEDEKIND SUMS IN FUNCTION FIELDS

We consider the Dedekind sum s(a, c) in rational function fields. It is very similar to the classical Dedekind sum D(a, c). The objective of this study is to give a good upper bound of the degree of the denominator of s(a, c). As an application of our result, we present a condition on the elements a1, a2, and c of 𝔽q[T] such that s(a1, c) = s(a2, c).

The Properties of Analogous to Dedekind Sums

The distribution problem of a sum analogous to Dedekind sums is studied by using the mean value theorem of the Dirichlet L-functions and the property of Dedekind sums , and an interesting mean square value formula is given.

On the mean value of dedekind sum weighted by the quadratic gauss sum

Various properties of classical Dedekind sums S(h, q) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H.Rademacher and E.Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.

Pruned Bit-Reversal Permutations: Mathematical Characterization, Fast Algorithms and Architectures

A mathematical characterization of serially pruned permutations (SPPs) employed in variable-length permuters and their associated fast pruning algorithms and architectures are proposed. Permuters are used in many signal processing systems for shuffling data and in communication systems as an adjunct to coding for error correction. Typically, only a small set of discrete permuter lengths are supported. Serial pruning is a simple technique to alter the length of a permutation to support a wider range of lengths, but results in a serial processing bottleneck. In this paper, parallelizing SPPs is formulated in terms of recursively computing sums involving integer floor functions using integer operations, in a fashion analogous to evaluating Dedekind sums. A mathematical treatment for bit-reversal permutations (BRPs) is presented, and closed-form expressions for BRP statistics including descents/ascents, major index, excedances/descedances, inversions, and serial correlations are derived. It is shown that BRP sequences have weak correlation properties. Moreover, a new statistic called permutation inliers that characterizes the pruning gap of pruned interleavers is proposed. Using this statistic, a recursive algorithm that computes the minimum inliers count of a pruned BR interleaver (PBRI) in logarithmic time is presented. This algorithm enables parallelizing a serial PBRI algorithm by any desired parallelism factor by computing the pruning gap in lookahead rather than a serial fashion, resulting in significant reduction in interleaving latency and memory overhead. Extensions to 2-D block and stream interleavers, as well as applications to pruned fast Fourier transforms and LTE turbo interleavers, are also presented. Moreover, hardware-efficient architectures for the proposed algorithms are developed. Simulation results of interleavers employed in modern communication standards demonstrate three to four orders of magnitude improvement in interleaving time compared to existing approaches.

Period functions and cotangent sums

We investigate the period function of $\sum_{n=1}^\infty\sigma_a(n)\e{nz}$, showing it can be analytically continued to $|\arg z|<\pi$ and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moments of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. In particular, we find a reciprocity formula for the Vasyunin sum.

Fast Pruned Interleaving

\boldmath In this paper, computationally efficient schemes for enumerating the so-called inliers of a wide range of permutations employed in pruned variable-size (turbo) interleavers are proposed. The objective is to accelerate pruned interleaving time in turbo codes by computing a statistic known as the pruning gap that enables determining a permuted address under pruning without serially permuting all its predecessors. It is shown that for any linear or quadratic permutation, including variations such as dithered relative prime or almost regular, the pruning gap can be computed in logarithmic time. Moreover, it is shown that Dedekind sums form efficient building blocks for enumerating inliers of the widely adopted polynomial-based permutations. An efficient algorithm for computing such sums in vector form using integer operations is presented. The results are extended to 2D and higher dimensional interleavers that combine multiple permutations along all dimensions, and closed-form expressions for inliers are derived. It is shown that the inliers statistic is a linear combination of the constituent permutation inliers. A lower bound on the minimum spread of serially pruned interleavers using the inliers statistic is also derived. Moreover, it is shown that serially pruned interleavers inherit the content-free property of the mother interleaver, and hence they are parallelizable. Simulation results of practical pruned turbo interleavers demonstrate a speedup improvement of several orders of magnitude compared to serial interleaving.

On the Hybrid Mean Value Involving Kloosterman Sums and Sums Analogous to Dedekind Sums

The main purpose of this paper is using the properties of Gauss sums and the mean value theorem of DirichletL-functions to study one kind of hybrid mean value problems involving Kloosterman sums and sums analogous to Dedekind sums and give two exact computational formulae for them.

An identity involving Dedekind sums and generalized Kloosterman sums

The various properties of classical Dedekind sums S(h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K(m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.

Heegaard–Floer homologies of (+1) surgeries on torus knots

We compute the Heegaard–Floer homology of $S^{3}_{1}(K)$ (the (+1) surgery on the torus knot T p,q ) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsváth–Szabó d-invariant. We relate the result to known knot invariants of T p,q as the genus and the Levine–Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard–Floer homologies of (+1) and (−1) surgeries on torus knots. This relation is best seen at the level of τ functions.

Rationality of Dedekind sums in function fields

In the previous paper we introduced the higher dimensional Dedekind sum in a function field (Bayad and Hamahata, Acta Arith. 152:71–80, 2012). The purpose of this paper is to present a criterion for the rationality of our Dedekind sum. To do so, we establish a connection between the field of definition of the Drinfeld module ϕ and the field of definition of the higher dimensional Dedekind sum sΛ(a0;a1,…,ad) associated to the A-lattice Λ, which corresponds to ϕ.

DEDEKIND SUMS WITH SMALL DENOMINATORS

Let (m, n) = 1 and S(m/n) = 12s(m/n), where s(m/n) is the usual Dedekind sum. Then [Formula: see text]. Let q ≥ 1 be a divisor of n. We give a necessary and sufficient condition for [Formula: see text] and, thereby, generalize a result of Rademacher that concerns the case q = 1. Further, we study the structure of possible denominators q of S(m/n). Finally, we show that for certain quadratic irrationals of odd period length l the convergents sk/tk, k ≡ l - 1 ( mod 2l), yield stationary values of S(sk/tk). This means that, for large values of k, the denominators q of these Dedekind sums are very small compared with tk.

Dedekind sums in function fields

Okada (J Number Theory, 130:1750–1762, 2010) introduced Dedekind sums associated to a certain A-lattice, and established the reciprocity law. In this paper, we introduce Dedekind sums for arbitrary A-lattice and establish the reciprocity law for them. We next introduce higher dimensional Dedekind sums for any A-lattice. These Dedekind sums are analogues of Zagier’s higher dimensional Dedekind sums. We discuss the reciprocity law, rationality and characterization of these sums.