Dedekind sums arising from newform Eisenstein series

For primitive nontrivial Dirichlet characters [Formula: see text] and [Formula: see text], we study the weight zero newform Eisenstein series [Formula: see text] at [Formula: see text]. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of [Formula: see text]. We also give a short proof of the reciprocity formula for this Dedekind sum.

A twisted generalization of the classical Dedekind sum

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.

On a recent reciprocity formula for Dedekind sums

Let [Formula: see text] denote the classical Dedekind sum and [Formula: see text]. Recently, Du and Zhang proved the following reciprocity formula. If [Formula: see text] and [Formula: see text] are odd natural numbers, [Formula: see text], then [Formula: see text] where [Formula: see text] and [Formula: see text]. 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 [Formula: see text]-series, our main tool is the three-term relation for Dedekind sums.

On the moduli of a Dedekind sum

Let [Formula: see text] denote the classical Dedekind sum and [Formula: see text]. Let [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], be the value of [Formula: see text]. In a previous paper, we showed that there are pairs [Formula: see text], [Formula: see text], such that [Formula: see text] for all [Formula: see text], the [Formula: see text]’s growing in [Formula: see text] exponentially. Here we exhibit such a sequence with [Formula: see text] a polynomial of degree [Formula: see text] in [Formula: see text].

Dedekind sums take each value infinitely many times

For [Formula: see text] and [Formula: see text], [Formula: see text], let [Formula: see text] denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs [Formula: see text], [Formula: see text], with [Formula: see text] tending to infinity as [Formula: see text] grows, such that [Formula: see text] for all [Formula: see text].

A distribution of rational homology 3-spheres captured by the CWL invariant Phase 1

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot [Formula: see text] with [Formula: see text].

An integer-valued expression of Dedekind sums

We prove a conjecture of Myerson and Phillips on when an expression involving Dedekind sums is an integer. We also provide a more general result and use this to extend the work of Myerson and Phillips studying whether the points of the graph of the one-variable Dedekind sum that fall on the line are dense.

The largest values of Dedekind sums

Let [Formula: see text] denote the classical Dedekind sum, where [Formula: see text] is a positive integer and [Formula: see text], [Formula: see text]. For a given positive integer [Formula: see text], we describe a set of at most [Formula: see text] numbers [Formula: see text] for which [Formula: see text] may be [Formula: see text], provided that [Formula: see text] is sufficiently large. For the numbers [Formula: see text] not in this set, [Formula: see text].