<abstract xmlns="http://www.w3.org/1999/xhtml"> For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , Iℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , Iℓ and d1, . . . , dℓ, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2. </abstract>

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , Iℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , Iℓ and d1, . . . , dℓ, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

On the Closure of the Image of the Generalized Divisor Function

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"On the Closure of the Image of the Generalized Divisor Function"}

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We...

On the Closure of the Image of the Generalized Divisor Function

Published Online: Jan 30, 2018

Page range: 77 - 90

Received: Nov 27, 2015

Accepted: Jan 10, 2017

DOI: https://doi.org/10.1515/udt-2017-0016

Keywords
Arithmetic functions, sum of divisors, topological closure, asymptotic densities

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the Closure of the Image of the Generalized Divisor Function

Published Online: Jan 30, 2018

Page range: 77 - 90

Received: Nov 27, 2015

Accepted: Jan 10, 2017

DOI: https://doi.org/10.1515/udt-2017-0016

KeywordsArithmetic functions, sum of divisors, topological closure, asymptotic densities

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Arithmetic functions, sum of divisors, topological closure, asymptotic densities