# Some properties and applications of non-trivial divisor functions

## Abstract

The *j*th 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 *j*th *non-trivial divisor function* \(c_j\), 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 function* \(c_j^{(r)}\), for \(r\ge 0\), 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.

## Keywords

Divisor functions Dirichlet series Hypergeometric series Arithmetic combinatorics## Mathematics Subject Classification

11A25 11A51 11M41 33C20 11B30## 1 Introduction

*j*th 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. In particular, \(d_2(n)\)—sometimes called the divisor function—counts the number of ordered pairs of positive integers whose product is

*n*, and therefore, considering only the first factor in each pair, also counts the number of divisors of

*n*(see papers 8 and 15 of [11] and p. 10 of [2]). 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

*x*, which is notoriously difficult if \(j \ge 3\), see e.g. [1, 8], and, for \(j=3\), [7].

In the present paper, we consider the rather less well-studied *j*th *non-trivial divisor function* \(c_j\), which counts the ordered proper factorisations of a positive integer into *j* factors, each of which is greater than or equal to 2. While \(d_j(n)\), for given *n*, is obviously monotone increasing in *j*, since factors of 1 can be freely introduced, \(c_j(n)\) will shrink back to 0 as *j* gets too large, and indeed \(c_j(n) = 0\) if \(n < 2^j\).

*associated divisor function*\(c_j^{(r)}\), for \(r \in \mathbb {N}_0\), by

The paper is organised as follows. In Sect. 2, after reviewing properties of \(d_j\), we proceed to study analogous properties of \(c_j\), specifically regarding its associated Dirichlet series and its representation in terms of binomial coefficient sums and hypergeometric series. A major complication in comparison to \(d_j\) arises from the fact that \(c_j\) is not multiplicative. We also provide formulae expressing \(c_j\) in terms of \(d_j\) and vice versa. We then introduce the associated divisor functions \(c_j^{(r)}\). Noting general inequalities between the three types of divisor function in Sect. 3, we 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 in Sect. 4 how they can be used to count principal reversible squares [10] and sum-and-distance systems of integers.

Throughout the paper, we use the notations \(\mathbb {N} = \{1, 2, 3, \ldots \}\), \(\mathbb {N}_0~=~\mathbb {N}~\cup ~\{0\}\). For *n* having prime factorisation \(n~=~ p_1^{a_1} p_2^{a_2} \cdots p_t^{a_t}\), we also use the symbol \(\varOmega (n) = \sum _{k=1}^t a_k\) .

## 2 Basic properties of standard, non-trivial and associated divisor functions

*j*-fold Dirichlet convolution of arithmetic functions \(f_1, f_2, \ldots , f_j\),

*e*is defined as

*n*shows that \(d_j\) is a multiplicative arithmetic function; however, it is not totally multiplicative, e.g. \(d_3(20) = 18 \ne 27 = d_3(2) d_3(10)\).

### Lemma 1

### Proof

*j*) For \(j=1\) the formula is trivial. Suppose \(j\in \mathbb {N}\) is such that (4) holds. Note that \(m | \prod _{i=1}^k p_i^{a_i}\) if and only if \(m = \prod _{i=1}^k p_i^{{\tilde{a}}_i}\) with \(0 \le {\tilde{a}}_i \le a_i\) for all \(i \in \{1, \ldots , k\}\). Using multi-index notation, we can write the latter condition in the form \(0 \le {\tilde{a}} \le a\). By (3),

### Remark 1

*j*-fold Dirichlet convolution \(c_j = (1 - e)^{*j}\). Hence it satisfies a slightly different sum-over-divisors recurrence relation compared to (3),

We emphasise that \(c_j\), unlike \(d_j\), is *not* a multiplicative arithmetic function. For example, \((2,5)=1\), and yet \(c_2(10)=2\ne 0 \times 0 = c_2(2) c_2(5).\)

In order to study the less symmetric multiplicative properties of \(c_j\), it is useful to express it in terms of its multiplicative cousin \(d_j\). When \(j=2\), the non-trivial divisors for any *n* are all the divisor except 1 and *n*, and hence \(c_2(n)=d_2(n)-2\) if \(n \ge 2\), and \(c_2(1) = d_2(1) - 1=0\). More generally, there is the following connection between the divisor function and the non-trivial divisor function.

### Lemma 2

### Proof

*j*-fold Dirichlet convolution,

*e*is the identity for the convolution product. The proof of the second identity is analogous, starting from \(d_j = ((1-e)+e)^{*j}\). \(\square \)

*a*, \(a^{\overline{m}} = (-1)^m\,(-a)!/(-a-m)!\) if \(-(a+m)\in \mathbb {N}_0\). By the usual convention on empty products, \(a^{\overline{0}} =1 \).

### Theorem 1

*n*has the prime factorisation \(n=p_1^{a_1}\ldots p_k^{a_k}\). Then the value of the non-trivial

*j*th divisor function at

*n*has the hypergeometric form

### Proof

### Lemma 3

*p*be a prime and \(j, a, b \in \mathbb {N}\). Then

### Proof

In analogy to the sum-over-divisors recurrence relation (3) for the divisor function \(d_j\), we define the *j*th associated divisor function \(c_j^{(r)} \) by the following recurrence.

### Definition 1

*r*, the

*associated divisor function*\(c_j^{(r)}\) is defined recursively by

This immediately gives the representation \(c_j^{(r)} = (1 - e)^{*j} * 1^{*r}\). Thus, we can interpret \(c_j^{(r)}(n)\) as the number of ordered factorisations of *n* into \(j+r\) factors, the first *j* of which are greater than 1. It follows that \(c_j^{(r)}(n) = 0\) for all \(r\in \mathbb {N}\) if \(n < 2^j\).

### Lemma 4

The following binomial form for the value of \(c_j^{(r)}\) at prime powers is somewhat analogous to Lemma 1, but note that the present function is not multiplicative.

### Lemma 5

*p*a prime. Then

### Proof

*f*by

*Gf*, so

*f*,

*g*, then

## 3 Ratios of divisor functions

The divisor function, non-trivial divisor function and associated divisor functions satisfy the following ordering relations.

### Lemma 6

### Proof

*n*|

*n*, so

Lemma 6 shows that, for any \(r\in \mathbb {N}_0\), the normalised divisor ratio function \(c_j^{(r)}/d_{j+r}\) takes rational values between 0 and 1, with the zeros occurring exactly when \(j>\varOmega (n)\). We have the following formulae for this function and the similar ratio \(c_j^{(r)}/d_r\).

### Theorem 2

### Proof

*n*has prime factorisation \(n = p_1^{a_1}\cdots p_t^{a_t}\).

*n*is a prime power. We note that in this case, Lemmata 1 and 5 give

## 4 Counting principal reversible squares

As an illustration for the use of the non-trivial and associated divisor functions, we show how they can be used to count the different principal reversible squares of a given size.

*reversible square matrix*\(M=\left( M_{i,j}\right) _{i,j \in \mathbb {Z}_n} \in \mathbb {R}^{n\times n}\) is an \(n \times n\) matrix with the following symmetry properties (cf. [10, 9]),

- (R)the row and column reversal symmetry$$\begin{aligned} M_{i,j} +M_{i,n+1-j}&=M_{i,k} +M_{i,n+1-k}, \\ M_{i,j}+M_{n+1-i,j}&=M_{k,j}+M_{n+1-k,j} \qquad (i,j,k \in \mathbb {Z}_n), \end{aligned}$$
- (V)
the vertex cross sum property \(M_{i,j}+M_{k,l}=M_{i,l}+M_{k,j}\) \((i,j,k,l \in \mathbb {Z}_n)\).

An \(n\times n\) *principal reversible square* is a reversible square matrix *M* such that \(\{M_{i,j}\mid i,j\in \mathbb {Z}_n\} = \{1,2, \ldots , n^2\}\), the entries in each row and each column appear in increasing order, and \(M_{1,j} = j\) (\(j\in \{1,2\}\)).

### Definition 2

*divisor path set*for

*n*(of length \(\alpha \)) if

### Theorem 3

Let \(n \in \mathbb {N}\). Then from any divisor path set for *n*, a unique \(n\times n\) principal reversible square can be constructed. Conversely, every \(n\times n\) principal reversible square arises from a unique divisor path set.

For the details of the construction and proof of Theorem 3, we refer the reader to Chap. 3 of [10]. In that book, a principal reversible square constructed from a divisor path set of length \(\alpha \) is said to have \(\alpha -1\) *progressive factors*.

*n*,

*n*), with the last entry omitted if \(j_\alpha = n.\)

Using the bijection between divisor path sets and principal reversible squares given by Theorem 3, we can count the number of different principal reversible squares of size \(n\times n\) in terms of the non-trivial and associated divisor functions of *n* as follows.

### Theorem 4

### Proof

By Theorem 3, it is sufficient to count the number of different divisor path sets for *n*.

*n*of length \(\alpha \). Then the left-hand tuple gives an ordered factorisation of

*n*into \(\alpha \) factors,

*n*into \(\alpha +1\) factors,

The statement of the theorem follows by summing over \(\alpha \in \mathbb {N}\), noting that \(c_\alpha (n) = 0\) if \(\alpha > \varOmega (n)\), and using Lemma 4 for the last identity. \(\square \)

### Remark 2

### Corollary 1

Let \(n\in \mathbb {N}\). Then \(N_n = 1\) if and only if *n* is prime.

### Proof

*n*is prime, then \(c_1(n) = 1\) and \(c_j(n) = 0\) for all \(j\ge 2\), and it follows that \(N_{n}=c_1(n)\left( c_1(n)+c_{2}(n)\right) =1\). Conversely, suppose \(n \ge 2\) is an integer such that \(N_n = 1\). By Theorem 4,

*n*is prime. \(\square \)

### Corollary 2

*p*. Then

### Proof

Counting principal reversible squares is of interest not only in view of their bijection to most perfect squares [10], but also because of their relationship with sum-and-distance systems. In the present context, these are composed of two finite component sets, of equal cardinality, of natural numbers, such that the numbers formed by considering all sums and all absolute differences of all pairs of numbers, each taken from one of the component sets, with or without inclusion of the component sets themselves, combine to an arithmetic progression without repetitions. Such systems arise naturally from the question of constructing a certain type of rank 2 traditional magic squares using the formulae given in [9]. We refer the reader to [6] for further details and for the extension of the following definitions and of Theorem 5 to any finite number of component sets of arbitrary finite cardinality.

### Definition 3

- (a)an \(m+m\)
*(non-inclusive) sum-and-distance system*if$$\begin{aligned} \{a_j+b_k, |a_j-b_k| : j,k\in \{1,\ldots ,m\}\} = \{1, 3, 5, \ldots , 4m^2-1\}; \end{aligned}$$ - (b)an \(m+m\)
*inclusive sum-and-distance system*if$$\begin{aligned} \{a_j, b_k, a_j+b_k, |a_j-b_k| : j,k\in \{1,\ldots ,m\}\} = \{1, 2, 3, \ldots , 2m(m+1)\}. \end{aligned}$$

### Example 1

*n*is even, \(n = 2m\). Then setting

Thus, we have proven the following statement.

### Theorem 5

Let \(m\in \mathbb {N}\). Then there is a bijection between the \(m+m\) non-inclusive sum-and-distance systems and the \(2m \times 2m\) principal reversible squares, and there is a bijection between the \(m+m\) inclusive sum-and-distance systems and the \((2m+1)\times (2m+1)\) principal reversible squares.

In conjunction with Theorem 4, this gives the following counting of non-inclusive and inclusive sum-and-distance systems.

### Corollary 3

To conclude, we briefly note that \(m+m\) sum-and-distance systems of either variety have the general property that the sum of squares of all entries of their component sets is invariant, determined only by the size *m*.

### Theorem 6

### Proof

## Notes

### Acknowledgements

The authors are grateful to the anonymous referee for constructive comments which helped streamline the paper, especially for suggesting the extensive use of convolution notation.

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