Abstract
The jth divisor function d_j, 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 functionc_j, 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 functionc_j^{(r)}, for rge 0, whose definition is motivated by the sum-over divisors recurrence for d_j. We give an overview of properties of d_j, c_j and c_j^{(r)}, 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.
Highlights
The jth divisor function d j, which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known arithmetic function
The divisor function lies at the heart of a number of open number theoretical problems, e.g. the additive divisor problem of finding the asymptotic of d j (n) d j (n + h) n≤x for large x, which is notoriously difficult if j ≥ 3, see e.g. [1,8], and, for j = 3, [7]
We introduce the associated divisor functions c(jr)
Summary
The jth divisor function d j , which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known arithmetic function. The non-trivial divisor function c j only counts ordered factorisations in which all factors are greater than 1, so by formula (2) it can be expressed as the j-fold Dirichlet convolution c j = (1 − e)∗ j. It satisfies a slightly different sum-over-divisors recurrence relation compared to (3), n c j+1(n) = (c j ∗ (1 − e))(n) = c j (m) (1 − e) m|n m c j (m) (n, j ∈ N). The following binomial form for the value of c(jr) at prime powers is somewhat analogous to Lemma 1, but note that the present function is not multiplicative.
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