The values of the Dedekind-Rademacher cocycle at real multiplication points

Heegaard Floer correction terms and Dedekind-Rademacher sums

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…

Dedekind-Rademacher上同调类的显式上循环与Darmon-Dasgupta测度

Explicit Cocycle of the Dedekind-Rademacher Cohomology Class and the Darmon-Dasgupta Measures

本文通过计算Dedekind-Rademacher测度值上同调类$μ_{DR}$的显式上循环代表元，明确其与Darmon-Dasgupta$p$-adic测度的关系，并解释了后者工作中的可选零次条件。

An analogue of the Rademacher function for generalized Dedekind sums in higher dimension

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 …

Mean values of Dedekind sums

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.

通过锥形zeta值的Dedekind和互反律

Reciprocity For Dedekind Sums via Conical Zeta Values

研究绝对连续函数相关Dedekind和的互反公式，推广经典Dedekind-Rademacher互反律，利用Fourier分析揭示多项式型函数互反性的锥形zeta值几何解释。

Rademacher-Carlitz Polynomials

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…

Rademacher符号的双曲类比

A Hyperbolic Analogue of the Rademacher Symbol

本文研究Rademacher符号的双曲类比，通过比较经典符号给出两类显式公式，并展示三类Eisenstein级数的Kronecker极限公式，得到权2的调和、极调和及局部调和Maass形式。

Transgressions of the Godbillon-Vey class and Rademacher functions

We construct, out of modular symbols, 1-traces that are invariant with respect to the actions of the Hopf algebra $\Hc_{1}$ on the crossed product $\Ac_{Q}$ of the algebra of modular forms of all levels by $\GL^+ (2,Q)$ investigated in earlier work. This provides a conceptual explanation for the construction of the Euler cocycle representing the image of the universal Godbillon-Vey class under the characteristic map of noncommutative Chern-Weil theory which we developed in our earlier work. We then refine the construction to produce secondary data by transgression. For the action determined by…

The simultaneous multiplication formulas for Apostol-Bernoulli polynomials and generalized Frobenius-Euler polynomials

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.

高阶 Rademacher 符号的整性

Integrality of the higher Rademacher symbols

本文证明 Duke 关于高阶 Rademacher 符号取整数值的猜想，借助 Bernoulli 数性质及 Kummer 同余的推广，阐明相关实二次域 zeta 值分母的简洁性。

Computing special values of partial zeta functions

We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $Ψ$, a group cocycle for $GL_{n} (\Z )$; the special values are computed as periods of $Ψ$, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.

Dedekind sums: a combinatorial-geometric viewpoint

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…