Classical Dedekind Sums Research Articles

Continued fractions and Hardy sums

The classical Dedekind sums s(d, c) can be represented as sums over the partial quotients of the continued fraction expansion of the rational dc. Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of θ-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial quotients of these continued fractions. As an application, we prove that the graph of the Hardy sums is dense in R×Z.

Algebraic properties of the values of newform Dedekind sums

We study the image of a generalized Dedekind sum relating to the weight zero Eisenstein series Eχ1,χ2. We show that the image is a lattice of full rank inside a number field determined by the characters χ1 and χ2. We also give a generalization of Knopp's identity for the classical Dedekind sum.

Reciprocity of poly-Dedekind-type DC sums involving poly-Euler functions

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 replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.

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.

SOME ARITHMETIC PROPERTIES OF AN ELLIPTIC DEDEKIND SUM

We give an explicit expression of the elliptic classical Dedekind sum which is a special case of multiple elliptic Dedekind sums introduced by Egami. We also determine the denominator of the rational part and zeros of the elliptic classical Dedekind sum.

An average of generalized Dedekind sums

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.

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].

The distribution of consecutive prime biases and sums of sawtooth random variables

AbstractIn recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

Characterization of Kollár surfaces

Koll\'ar introduced in [Ko08] the surfaces $$(x_1^{a_1}x_2+x_2^{a_2}x_3+x_3^{a_3}x_4+x_4^{a_4}x_1=0)\subset \mathbb{P}(w_1,w_2,w_3,w_4)$$ where $w_i=W_i/w^*$, $W_i=a_{i+1}a_{i+2}a_{i+3}-a_{i+2}a_{i+3}+a_{i+3}-1$, and $w^*=$gcd$(W_1,\ldots,W_4)$. The aim was to give many interesting examples of $\mathbb{Q}$-homology projective planes. They occur when $w^*=1$. For that case, we prove that Koll\'ar surfaces are Hwang-Keum [HK12] surfaces. For $w^*>1$, we construct a geometrically explicit birational map between Koll\'ar surfaces and cyclic covers $z^{w^*}=l_1^{a_2 a_3 a_4} l_2^{-a_3 a_4} l_3^{a_4} l_4^{-1}$, where $\{l_1,l_2,l_3,l_4\}$ are four general lines in $\mathbb{P}^2$. In addition, by using various properties on classical Dedekind sums, we prove that: (a) For any $w^*>1$, we have $p_g=0$ iff the Koll\'ar surface is rational. This happens when $a_{i+1} \equiv 1$ or $a_{i}a_{i+1} \equiv -1 ($mod $w^*)$ for some $i$. (b) For any $w^*>1$, we have $p_g=1$ iff the Koll\'ar surface is birational to a K3 surface. We classify this situation. (c) For $w^*>>0$, we have that the smooth minimal model $S$ of a generic Koll\'ar surface is of general type with $K_{S}^2/e(S) \to 1$.

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].

On the values of Dedekind sums

Let s(a,b) denote the classical Dedekind sum and S(a,b)=12s(a,b). For a given denominator q∈N, we study the numerators k∈Z of the values k/q, (k,q)=1, of Dedekind sums S(a,b). Our main result says that if k is such a numerator, then the whole residue class of k modulo (q2−1)q consists of numerators of this kind. This fact reduces the task of finding all possible numerators k to that of finding representatives for finitely many residue classes modulo (q2−1)q. By means of the proof of this result we have determined all possible numerators k for 2≤q≤60, the case q=1 being trivial. The result of this search suggests a conjecture about all possible values k/q, (k,q)=1, of Dedekind sums S(a,b) for an arbitrary q∈N.

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].

Dedekind sums and the transformation formula in function fields

Dedekind used the classical Dedekind sum [Formula: see text] to describe the transformation of [Formula: see text] under the substitution [Formula: see text]. In this paper, we use the Dedekind sum [Formula: see text] in function fields to describe the transformation of a certain series under the substitution [Formula: see text].

On the Dedekind sums and two-term exponential sums

In this paper, the authors use the analytic methods and the properties of character sums mod p to study the computational problem of one kind of mean value involving the classical Dedekind sums and two-term exponential sums, and give an exact computational formula for it.

Approximation of rational numbers by Dedekind sums

Given a rational number x and a bound ε, we exhibit m, n such that |x - s(m, n)| < ε. Here s(m, n) is the classical Dedekind sum and the parameters m and n are completely explicit in terms of x and ε.

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).

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.

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 optimal upper bound on the number of vanishing correction terms of lens spaces with square order second cohomology.