Dedekind's Function Research Articles

On a modification of $\underline{Set}(n)$

A modification of the set $\underline{\rm Set}(n)$ for a fixed natural number $n$ is introduced in the form: $\underline{\rm Set}(n, f)$, where $f$ is an arithmetic function. The sets $\underline{\rm Set}(n,\varphi), \underline{\rm Set}(n,\psi), \underline{\rm Set}(n,\sigma)$ are discussed, where $\varphi, \psi$ and $\sigma$ are Euler's function, Dedekind's function and the sum of the positive divisors of $n$, respectively.

On sums of gcd-sum functions

Let [Formula: see text] denote the greatest common divisor of integers [Formula: see text] and [Formula: see text]. Define [Formula: see text] for any arithmetical function [Formula: see text]. We shall derive several asymptotic expansions of the error terms [Formula: see text] for the partial sums of [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where [Formula: see text] and [Formula: see text] are the Euler totient function and the Dedekind function, respectively. Furthermore, we establish several formulas concerning the sum function [Formula: see text] and deduce several related results.

Sums of Averages of GCD-Sum Functions II

Let gcd (k,j) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define Mr(x;f):=∑k≤x1kr+1∑j=1kjrf(gcd(j,k))\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_r(x; f) := \\sum _{k\\le x}\\frac{1}{k^{r+1}}\\sum _{j=1}^{k}j^{r}f(\\gcd (j,k)) \\end{aligned}$$\\end{document}for any large real number xge 5, where f is any arithmetical function. Let phi , and psi denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of M_r(x; mathrm{id}), M_r(x;{phi }) and M_r(x;{psi }). Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of M_r(x;mathrm{id}) for any large positive number x>5 satisfying x=[x]+frac{1}{2}.

A note on some congruences involving arithmetic functions

We consider some congruences involving arithmetical functions. For example, we study the congruences nψ(n) ≡ 2 (mod ϕ(n)), nϕ(n) ≡ 2 (mod ψ(n)), ψ(n)d(n) − 2 ≡ 0 (mod n), where ϕ(n), ψ(n), d(n) denote Euler's totient, Dedekind's function, and the number of divisors of n, respectively. Two duals of the Lehmer congruence n − 1 ≡ 0 (mod ϕ(n)) are also considered.

Sums of averages of gcd-sum functions

Let gcd⁡(k,j) be the greatest common divisor of the integers k and j. We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is ∑k≤x1kr+1∑j=1kjrf(gcd⁡(k,j)) with f=id,ϕ,ϕs,ψ and ψs for any fixed positive integers r and s, where ϕ,ϕs,ψ and ψs are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the error term of the gcd-sum functions ∑k≤x1kr+1⋅∑j=1kjrϕ(gcd⁡(k,j)) and ∑k≤x1kr+1∑j=1kjrψ(gcd⁡(k,j)).

Lower estimates related to the twisted Dedekind function

Let [Formula: see text] be a Dirichlet character. The main goal of this paper is to study oscillations of the difference [Formula: see text] where [Formula: see text] denotes the twisted Dedekind function. We prove that for infinitely many odd characters [Formula: see text] called “good”, we have [Formula: see text], and [Formula: see text] when [Formula: see text] is real. We give a necessary and sufficient condition for [Formula: see text] to be good, and in particular we prove that all odd primitive characters are good. We show also that there are infinitely many moduli [Formula: see text], including all prime powers [Formula: see text], for which all odd characters [Formula: see text] are good.

Generalised Weber functions

A generalised Weber function is given by wN(z) = η(z/N)/η(z), where η(z) is the Dedekind function and N is any integer; the original function corresponds to N = 2. We classify the cases where some power w e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating wN(z) and j(z).

Extremal orders of some functions connected to regular integers modulo n

Abstract Let V (n) denote the number of positive regular integers (mod n) less than or equal to n. We give extremal orders of , , , , where σ(n), ψ(n) are the sum-of-divisors function and the Dedekind function, respectively. We also give extremal orders for and , where σ*(n) and Φ*(n) represent the sum of the unitary divisors of n and the unitary function corresponding to Φ(n), the Euler's function. Finally, we study some extremal orders of compositions f(g(n)), involving the functions from above.

Tomographically complete sets of orthonormal bases in finite systems

Quantum systems where the position and momentum are in the ring (d is an odd integer) are considered. A tomographically complete set of bases, i.e. a set of bases such that probabilities from tomography experiments corresponding to these bases, can be used for the calculation of an arbitrary density matrix, is considered. Such a set of bases is constructed by considering a factorization of a system with dimension d = p1p2 (where p1, p2 are prime numbers) in terms of two subsystems with dimensions p1 and p2. Appropriate combination of the mutually unbiased bases in the subsystems, leads to a tomographically complete set of bases in the full system. This set leads to probabilities along all the lines through the origin (0, 0) in the phase space, with exactly d points. It is shown that the number of these lines is ψ(d) (the Dedekind function). The general theory is exemplified with examples.

Values of arithmetical functions equal to a sum of two squares

Let φ(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n ≤ x for which φ(n) is the sum of two square numbers. We also obtain similar results for the Dedekind function ψ(n) and the sum of divisors function σ(n).

Nonrelativistic fermions in magnetic fields: a quantum field theory approach

The statistical mechanics of nonrelativistic fermions in a constant magnetic field is considered from the quantum field theory point of view. The fermionic determinant is computed using a general procedure that contains all possible regularizations. The nonrelativistic grand-potential can be expressed in terms polylogarithm functions, whereas the partition function in 2+1 dimensions and vanishing chemical potential can be compactly written in terms of the Dedekind eta function. The strong and weak magnetic fields limits are easily studied in the latter case by using the duality properties of the Dedekind function.

Some combinatorics of imperfect information

We can use the compositional semantics of Hodges [9] to show that any compositional semantics for logics of imperfect information must obey certain constraints on the number of semantically inequivalent formulas. As a corollary, there is no compositional semantics for the ‘independence-friendly’ logic of Hintikka and Sandu (henceforth IF) in which the interpretation in a structure A of each 1 -ary formula is a subset of the domain of A (Corollary 6.2 below proves this and more). After a fashion, this rescues a claim of Hintikka and provides the proof which he lacked:… there is no realistic hope of formulating compositional truth-conditions for [sentences of IF], even though I have not given a strict impossibility proof to that effect.(Hintikka [6] page 110ff.) One curious spinoff is that there is a structure of cardinality 6 on which the logic of Hintikka and Sandu gives nearly eight million inequivalent formulas in one free variable (which is more than the population of Finland).We thank the referee for a sensible change of notation, and Joel Berman and Stan Burris for bringing us up to date with the computation of Dedekind's function (see section 4). Our own calculations, utterly trivial by comparison, were done with Maple V.The paper Hodges [9] (cf. [10]) gave a compositional semantics for a language with some devices of imperfect information. The language was complicated, because it allowed imperfect information both at quantifiers and at conjunctions and disjunctions.

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.

Eine Anwendung von η(τ)

Abelian differentials of four elliptic function fields may be constructed from Dedekind's function η. Each of the fields mentioned belongs to a congruence subgroup of index 12 of the modular group. A method involving Weierstrass' ℘-functions then yields the respective absolute invariants. Two of the fields are thus seen not to admit of complex multiplication.

On product expansions of theta-functions

In [1], Hecke set up a general method of passing back and forth between Dirichlet series and functions on the upper half plane (via the Mellin transform) and showed how functional equations on the one side correspond to functional equations on the other. Weil [21 has remarked that this method is applicable in a more general setting than in [1], and has shown in this manner that the functional equation for Dedekind's function fl(r) is an immediate consequence of the functional equation for the Riemann zeta-function. In this note we give a proof along these lines of the product expansion of the basic theta-function a-z=3 =t6. We start with