Dedekind Sums Research Articles

Dedekind sums and Fourier coefficients of modular forms

We prove the following result on the distribution of Dedekind sums: lim M→∞ logM M ∑ c=1 M 1 c ∑ D modc ∗ g S(d,c), d c = 12 π 2 ∫ −∞ ∞ ∫ R Z g(x,y) dy dx, for each compactly supported continuous function g on R × (R/Z). The proof uses Kuznetsov's sum formula in the modular case for varying real weight.

On Rademacher's three-term relations for Dedekind sums

On Rademacher's three-term relations for Dedekind sums

On the reciprocity formula for generalized Dedekind sums

On the reciprocity formula for generalized Dedekind sums

On Dedekind sums and analogs

On Dedekind sums and analogs

P-adic interpolation of dedekind sums

In this article we give an explicit representation of p-adic Dedekind sums and their reciprocity laws by using p-adic measure theory. We then study the consequences of the p-adic reciprocity law for particular positive integer values in which case we can recover a reciprocity law for Dedekind sums attached to particular Dirichlet characters. This gives a proof different from that of Nagasaka.

Dedekind sums and uniform distribution

Let s( d, c) be the Dedekind sum; let r be a nonzero real number. We show that the sequence of points in two-space given by 〈 ( d c , rs(d,c)) 〉, c = 1, 2,…, 0 < d < c, ( d, c) = 1, is uniformly distributed (mod 1).

A relation between Dedekind sums and Kloosterman sums

A relation between Dedekind sums and Kloosterman sums

Three-term relations for Hardy sums

L. A. Goldberg (thesis, Univ. of Illinois, Urbana, 1981) discovered some three-term and mixed three-term relations for Hardy sums. His proofs are based on Berndt's transformation formulae for the logarithms of the classical theta functions. In this paper, we give elementary proofs for all of Goldberg's results and also prove some new three-term relations for Dedekind sums.

Some arithmetic on Dedekind sums

Some arithmetic on Dedekind sums

Some new reciprocity formulas for generalized Dedekind sums

Some new reciprocity formulas for generalized Dedekind sums

An Application of the Reciprocity Theorem for Dedekind Sums

An Application of the Reciprocity Theorem for Dedekind Sums

Cotangent sums, a further generalization of Dedekind sums

In this article a simple proof for a reciprocity formula for sums of cotangent functions is presented. If the second arguments are zero, these sums are four times the original Dedekind sums. Subsequently, explicit expressions are derived. Finally, it is shown that the Berndt sums are special cases of these sums.

Analytic Properties of Arithmetic Sums Arising in the Theory of the Classical Theta-Functions

In the transformation formulae for the logarithms of the classical theta-functions, there arise certain arithmetic sums that are analogous to Dedekind sums. In this paper, analytic properties of these arithmetic sums are established. In particular, reciprocity theorems are proved and representations as finite trigonometric sums are given. Moreover, certain infinite series and certain doubly infinite series are evaluated in closed form in terms of these arithmetic sums.

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series G p(x) = Σ n−1 ∞ n −px n (1−x n) relating G p ( e 2 πiτ ) to G p ( e 2 πiAτ ), where p > 1 is an odd integer and Aτ = (aτ + b) (cτ + d) is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function log η(τ) = πiτ 12 − G 1(e 2πiτ) , and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.

Elementary proofs of Berndt’s reciprocity laws

Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz.

KRONECKER'S LIMIT FORMULA IN A REAL QUADRATIC FIELD

In this paper the author obtains a formula representing the free term of the Laurent expansion of the zeta functions of absolute and ray classes of a real quadratic field at the point 1, as well as the value of Hecke's L-function corresponding to the signature character. The formula contains Dedekind sums and sums analogous to them in which functions depending on the same arguments as in ordinary Dedekind sums appear. Bibliography: 4 titles.

An elementary proof of the Petersson-Knopp theorem on Dedekind sums

A brief and elementary proof of Petersson and Knopp's recent theorem on Dedekind sums is given. The proof is based on a result of Subrahmanyam.

Dedekind sums and Hecke operators

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)

Hecke operators and an identity for the dedekind sums

If h, k ∈ Z, k > 0, the Dedekind sum is given by s(h,k) = ∑ μ=1 k μ k hμ k , with ((x)) = x − [x] − 1 2 , x∉Z , =0 , x∈Z . The Hecke operators T n for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities ( n ∈ Z +) ∑ ∑ s(ah+bk,dk) = σ(n)s(h,k) , ad=n b( mod d) d>0 where ( h, k) = 1, k > 0 and σ( n) is the sum of the positive divisors of n. Petersson had earlier proved (∗) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (∗) when n is prime.

On generalized Dedekind sums

In this paper we give a simple proof for the reciprocity formula for the generalized Dedekind sums and derive an explicit expression for these sums.