Some properties and applications of non-trivial divisor functions

Lettington, Matthew, Schmidt, Karl, Huxley, Martin Neil and Hill, Sally 2019. Some properties and applications of non-trivial divisor functions. Ramanujan Journal 10.1007/s11139-018-0093-9

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Abstract

The jth divisor function dj , which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function cj , which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor function c(r)j , for r≥0 , whose definition is motivated by the sum-over divisors recurrence for dj . We give an overview of properties of dj , cj and c(r)j , specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers.

Item Type:	Article
Date Type:	Published Online
Status:	In Press
Schools:	Mathematics
Publisher:	Springer Verlag (Germany)
ISSN:	1382-4090
Funders:	EPSRC
Date of First Compliant Deposit:	22 September 2018
Date of Acceptance:	17 September 2018
Last Modified:	29 Jan 2020 01:01
URI:	http://orca.cf.ac.uk/id/eprint/115177

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