Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

1 Definition
2 Example
3 Table of values
4 Properties
5 Series relations
6 Approximate growth rate
7 See also
8 Notes
9 References

Definition

The sum of positive divisors function σ_x(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n, or

$\sigma_{x}(n)=\sum_{d|n} d^x\,\! .$

The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ₀(n), or the number-of-divisors function^[1]^[2] (sequence A000005 in OEIS). When x is 1, the function is called the sigma function or sum-of-divisors function,^[3]^[4] and the subscript is often omitted, so σ(n) is equivalent to σ₁(n) ( A000203). The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, A001065), and equals σ₁(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

Example

For example, σ₀(12) is the number of the divisors of 12:

$\begin{align} \sigma_{0}(12) & = 1^0 %2B 2^0 %2B 3^0 %2B 4^0 %2B 6^0 %2B 12^0 \\ & = 1 %2B 1 %2B 1 %2B 1 %2B 1 %2B 1 = 6, \end{align}$

while σ₁(12) is the sum of all the divisors:

$\begin{align} \sigma_{1}(12) & = 1^1 %2B 2^1 %2B 3^1 %2B 4^1 %2B 6^1 %2B 12^1 \\ & = 1 %2B 2 %2B 3 %2B 4 %2B 6 %2B 12 = 28, \end{align}$

and the aliquot sum s(12) of proper divisors is:

$\begin{align} s(12) & = 1^1 %2B 2^1 %2B 3^1 %2B 4^1 %2B 6^1 \\ & = 1 %2B 2 %2B 3 %2B 4 %2B 6 = 16. \end{align}$

Table of values

n	Divisors	σ₀(n)	σ₁(n)	s(n) = σ₁(n) − n	Comment
1	1	1	1	0	square number: σ₀(n) is odd; power of 2: s(n) = n − 1 (almost-perfect)
2	1,2	2	3	1
3	1,3	2	4	1
4	1,2,4	3	7	3	square number: σ₀(n) is odd; power of 2: s(n) = n − 1 (almost-perfect)
5	1,5	2	6	1
6	1,2,3,6	4	12	6	first perfect number: s(n) = n
7	1,7	2	8	1
8	1,2,4,8	4	15	7	power of 2: s(n) = n − 1 (almost-perfect)
9	1,3,9	3	13	4	square number: σ₀(n) is odd
10	1,2,5,10	4	18	8
11	1,11	2	12	1
12	1,2,3,4,6,12	6	28	16	first abundant number: s(n) > n
13	1,13	2	14	1
14	1,2,7,14	4	24	10
15	1,3,5,15	4	24	9
16	1,2,4,8,16	5	31	15	square number: σ₀(n) is odd; power of 2: s(n) = n − 1 (almost-perfect)

The cases $x=2$ , $x=3$ and so on are tabulated in A001157, A001158, A001159, A001160, A013954, A013955 ...

Properties

For a non-square integer every divisor d of n is paired with divisor n/d of n and $\sigma_{0}(n)$ is then even; for a square integer one divisor (namely $\sqrt n$ ) is not paired with a distinct divisor and $\sigma_{0}(n)$ is then odd.

For a prime number p,

$\begin{align} d(p) & = 2 \\ d(p^n) & = n%2B1 \\ \sigma(p) & = p%2B1 \end{align}$

because by definition, the factors of a prime number are 1 and itself. Also,where p_n# denotes the primorial,

$d(p_n\#) = 2^n \,$

since n prime factors allow a sequence of binary selection ( $p_{i}$ or 1) from n terms for each proper divisor formed.

Clearly, 1 < d(n) < n and σ(n) > n for all n > 2.

The divisor function is multiplicative, but not completely multiplicative. The consequence of this is that, if we write

$n = \prod_{i=1}^r p_i^{a_i}$

where r = ω(n) is the number of distinct prime factors of n, p_i is the ith prime factor, and a_i is the maximum power of p_i by which n is divisible, then we have

$\sigma_x(n) = \prod_{i=1}^{r} \frac{p_{i}^{(a_{i}%2B1)x}-1}{p_{i}^x-1}$

which is equivalent to the useful formula:

$\sigma_x(n) = \prod_{i=1}^r \sum_{j=0}^{a_i} p_i^{j x} = \prod_{i=1}^r (1 %2B p_i^x %2B p_i^{2x} %2B \cdots %2B p_i^{a_i x}).$

It follows (by setting x = 0) that d(n) is:

$d(n)=\prod_{i=1}^r (a_i%2B1).$

For example, if n is 24, there are two prime factors (p₁ is 2; p₂ is 3); noting that 24 is the product of 2³×3¹, a₁ is 3 and a₂ is 1. Thus we can calculate d(24) as so:

$\begin{align} d(24) & = \prod_{i=1}^{2} (a_i%2B1) \\ & = (3 %2B 1)(1 %2B 1) = 4 \times 2 = 8. \end{align}$

The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.

We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, i.e. the divisors of n excluding n itself. This function is the one used to recognize perfect numbers which are the n for which s(n) = n. If s(n) > n then n is an abundant number and if s(n) < n then n is a deficient number.

If n is a power of 2, e.g. $n = 2^k$ , then $\sigma(n) = 2.2^k - 1 = 2n - 1,$ and s(n) = n - 1, which makes n almost-perfect.

As an example, for two distinct primes p and q with p < q, let

$n = pq. \,$

Then

$\sigma(n) = (p%2B1)(q%2B1) = n %2B 1 %2B (p%2Bq), \,$

$\phi(n) = (p-1)(q-1) = n %2B 1 - (p%2Bq), \,$

and

$n %2B 1 = (\sigma(n) %2B \phi(n))/2, \,$

$p %2B q = (\sigma(n) - \phi(n))/2, \,$

where φ(n) is Euler's totient function.

Then, the roots of:

$(x-p)(x-q) = x^2 - (p%2Bq)x %2B n = x^2 - [(\sigma(n) - \phi(n))/2]x %2B [(\sigma(n) %2B \phi(n))/2 - 1] = 0 \,$

allows us to express p and q in terms of σ(n) and φ(n) only, without even knowing n or p+q, as:

$p = (\sigma(n) - \phi(n))/4 - \sqrt{[(\sigma(n) - \phi(n))/4]^2 - [(\sigma(n) %2B \phi(n))/2 - 1]}, \,$

$q = (\sigma(n) - \phi(n))/4 %2B \sqrt{[(\sigma(n) - \phi(n))/4]^2 - [(\sigma(n) %2B \phi(n))/2 - 1]}. \,$

Also, knowing n and either σ(n) or φ(n) (or knowing p+q and either σ(n) or φ(n)) allows us to easily find p and q.

In 1984, Roger Heath-Brown proved that

d(n) = d(n + 1)

will occur infinitely often.

Series relations

Two Dirichlet series involving the divisor function are:

$\sum_{n=1}^\infty \frac{\sigma_{a}(n)}{n^s} = \zeta(s) \zeta(s-a),$

which for d(n) = σ₀(n) gives

$\sum_{n=1}^\infty \frac{d(n)}{n^s} = \zeta^2(s),$

and

$\sum_{n=1}^\infty \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s) \zeta(s-a) \zeta(s-b) \zeta(s-a-b)}{\zeta(2s-a-b)}.$

A Lambert series involving the divisor function is:

$\sum_{n=1}^\infty q^n \sigma_a(n) = \sum_{n=1}^\infty \frac{n^a q^n}{1-q^n}$

for arbitrary complex |q| ≤ 1 and a. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.

Approximate growth rate

In little-o notation, the divisor function satisfies the inequality (see page 296 of Apostol’s book^[5])

$\mbox{for all }\epsilon>0,\quad d(n)=o(n^\epsilon).$

More precisely, Severin Wigert showed that

$\limsup_{n\to\infty}\frac{\log d(n)}{\log n/\log\log n}=\log2.$

On the other hand, since there are infinitely many prime numbers,

$\liminf_{n\to\infty} d(n)=2.$

In Big-O notation, Dirichlet showed that the average order of the divisor function satisfies the following inequality (see Theorem 3.3 of Apostol’s book^[5])

$\mbox{for all } x\geq1, \sum_{n\leq x}d(n)=x\log x%2B(2\gamma-1)x%2BO(\sqrt{x}),$

where $\gamma$ is Euler's constant. Improving the bound $O(\sqrt{x})$ in this formula is known as Dirichlet's divisor problem

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by:

$\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma,$

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that

$\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^{\gamma},$

where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality:

$\ \sigma(n) < e^\gamma n \log \log n$ (Robin's inequality)

holds for all sufficiently large n. In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984). This is Robin's theorem and the inequality became known after him. The largest known value that violates the inequality is n=5,040. If the Riemann hypothesis is true, there are no greater exceptions. If the hypothesis is false, then Robin showed there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n ≥ 5,041 must be superabundant (Akbary & Friggstad 2009). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime (Choie et al. 2007).

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that

$\sigma(n) \le H_n %2B \ln(H_n)e^{H_n}$

for every natural number n, where $H_n$ is the nth harmonic number, (Lagarias 2002).

Robin also proved, unconditionally, that the inequality

$\ \sigma(n) < e^\gamma n \log \log n %2B \frac{0.6483\ n}{\log \log n}$

holds for all n ≥ 3.

Notes

^ Long (1972, p. 46)
^ Pettofrezzo & Byrkit (1970, p. 63)
^ Long (1972, p. 46)
^ Pettofrezzo & Byrkit (1970, p. 58)
^ ^a ^b Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929

References

Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis", American Mathematical Monthly 116 (3): 273–275, doi:10.4169/193009709X470128, http://webdocs.cs.ualberta.ca/~zacharyf/Papers/superabundant.pdf .
Bach, Eric; Shallit, Jeffrey, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN 0-262-02405-5, see page 234 in section 8.8.
Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011), "Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis", INTEGERS: the Electronic Journal of Combinatorial Number Theory 11: A33, http://www.integers-ejcnt.org/l33/l33.pdf
Choie, YoungJu; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), "On Robin's criterion for the Riemann hypothesis", Journal de théorie des nombres de Bordeaux 19 (2): 357–372, arXiv:math.NT/0604314, doi:10.5802/jtnb.591, ISSN 1246-7405, MR 2394891, http://jtnb.cedram.org/item?id=JTNB_2007__19_2_357_0
Grönwall, Thomas Hakon (1913), "Some asymptotic expressions in the theory of numbers", Transactions of the American Mathematical Society 14: 113–122, doi:10.1090/S0002-9947-1913-1500940-6
Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly 109 (6): 534–543, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008
Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company
Nicolas, Jean-Louis; Ramanujan, Srinivasa; Robin, Guy (1997), "Highly composite numbers", The Ramanujan Journal. an International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan 1 (2): 119–153, doi:10.1023/A:1009764017495, ISSN 1382-4090, MR 1606180
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall
Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées. Neuvième Série 63 (2): 187–213, ISSN 0021-7824, MR 774171
Weisstein, Eric W., "Divisor Function" from MathWorld.
Weisstein, Eric W., "Robin's Theorem" from MathWorld.
Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.

Divisibility-based sets of integers

Overview	Integer factorization · Divisor · Unitary divisor · Divisor function · Prime factor · Fundamental theorem of arithmetic

Forms of factorization	Prime number · Composite number · Semiprime number · Pronic number · Sphenic number · Square-free number · Powerful number · Perfect power · Achilles number · Smooth number · Regular number · Rough number · Unusual number

Constrained divisor sums	Perfect number · Almost perfect number · Quasiperfect number · Multiply perfect number · Hyperperfect number · Superperfect number · Unitary perfect number · Semiperfect number · Primitive semiperfect number · Practical number

Numbers with many divisors	Abundant number · Primitive abundant number · Highly abundant number · Superabundant number · Colossally abundant number · Highly composite number · Superior highly composite number · Weird number

Numbers related to aliquot sequences	Untouchable number · Amicable number · Sociable number

Other	Deficient number · Friendly number · Solitary number · Sublime number · Harmonic divisor number · Frugal number · Equidigital number · Extravagant number