Divisor function

Divisor function σ₀(n) up to n = 250

Sigma function σ₁(n) up to n = 250

Sum of the squares of divisors, σ₂(n), up to n = 250

Sum of cubes of divisors, σ₃(n) up to n = 250

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). 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; these are treated separately in the article Ramanujan's sum.

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

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. It can be expressed in sigma notation as

\sigma _{x}(n)=\sum _{d\mid n}d^{x}\,\!,

where ${d\mid n}$ is shorthand for "d divides n". 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] ( A000005). When x is 1, the function is called the sigma function or sum-of-divisors function,^[1]^[3] and the subscript is often omitted, so σ(n) is the same as 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 + 2^0 + 3^0 + 4^0 + 6^0 + 12^0 \\ & = 1 + 1 + 1 + 1 + 1 + 1 = 6, \end{align}

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

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

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

\begin{align} s(12) & = 1^1 + 2^1 + 3^1 + 4^1 + 6^1 \\ & = 1 + 2 + 3 + 4 + 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	Prime: σ₁(n) = 1 + n so s(n) = 1; power of 2: s(n) = n − 1 (almost-perfect)
3	1, 3	2	4	1	Prime: σ₁(n) = 1 + n so s(n) = 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	Prime: σ₁(n) = 1 + n so s(n) = 1
6	1, 2, 3, 6	4	12	6	first perfect number: s(n) = n
7	1, 7	2	8	1	Prime: σ₁(n) = 1 + n so s(n) = 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	Prime: σ₁(n) = 1 + n so s(n) = 1
12	1, 2, 3, 4, 6, 12	6	28	16	first abundant number: s(n) > n
13	1, 13	2	14	1	Prime: σ₁(n) = 1 + n so s(n) = 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)

σ₀(n)	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0+		1	2	2	3	2	4	2	4	3	4	2
12+	6	2	4	4	5	2	6	2	6	4	4	2
24+	8	3	4	4	6	2	8	2	6	4	4	4
36+	9	2	4	4	8	2	8	2	6	6	4	2
48+	10	3	6	4	6	2	8	4	8	4	4	2
60+	12	2	4	6	7	4	8	2	6	4	8	2
72+	12	2	4	6	6	4	8	2	10	5	4	2
84+	12	4	4	4	8	2	12	4	6	4	4	4
96+	12	2	6	6	9	2	8	2	8	8	4	2
108+	12	2	8	4	10	2	8	4	6	6	4	4
120+	16	3	4	4	6	4	12	2	8	4	8	2
132+	12	4	4	8	8	2	8	2	12	4	4	4

σ₁(n)	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0+		1	3	4	7	6	12	8	15	13	18	12
12+	28	14	24	24	31	18	39	20	42	32	36	24
24+	60	31	42	40	56	30	72	32	63	48	54	48
36+	91	38	60	56	90	42	96	44	84	78	72	48
48+	124	57	93	72	98	54	120	72	120	80	90	60
60+	168	62	96	104	127	84	144	68	126	96	144	72
72+	195	74	114	124	140	96	168	80	186	121	126	84
84+	224	108	132	120	180	90	234	112	168	128	144	120
96+	252	98	171	156	217	102	216	104	210	192	162	108
108+	280	110	216	152	248	114	240	144	210	182	180	144
120+	360	133	186	168	224	156	312	128	255	176	252	132
132+	336	160	204	240	270	138	288	140	336	192	216	168

σ₂(n)	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0+		1	5	10	21	26	50	50	85	91	130	122
12+	210	170	250	260	341	290	455	362	546	500	610	530
24+	850	651	850	820	1050	842	1300	962	1365	1220	1450	1300
36+	1911	1370	1810	1700	2210	1682	2500	1850	2562	2366	2650	2210
48+	3410	2451	3255	2900	3570	2810	4100	3172	4250	3620	4210	3482
60+	5460	3722	4810	4550	5461	4420	6100	4490	6090	5300	6500	5042
72+	7735	5330	6850	6510	7602	6100	8500	6242	8866	7381	8410	6890
84+	10500	7540	9250	8420	10370	7922	11830	8500	11130	9620	11050	9412
96+	13650	9410	12255	11102	13671	10202	14500	10610	14450	13000	14050	11450
108+	17220	11882	15860	13700	17050	12770	18100	13780	17682	15470	17410	14500
120+	22100	14763	18610	16820	20202	16276	22750	16130	21845	18500	22100	17162
132+	25620	18100	22450	21320	24650	18770	26500	19322	27300	22100	25210	20740

The cases x = 2 to 5 are listed in A001157 − A001160, x = 6 to 24 are listed in A013954 − A013972.

Properties

Formulas at prime powers

For a prime number p,

{\begin{aligned}\sigma _{0}(p)&=2\\\sigma _{0}(p^{n})&=n+1\\\sigma _{1}(p)&=p+1\end{aligned}}

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

\sigma_0(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, $1 < \sigma_0(n) < n$ 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}+1)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 + p_i^x + p_i^{2x} + \cdots + p_i^{a_i x}).

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

\sigma_0(n)=\prod_{i=1}^r (a_i+1).

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 $\sigma_0(24)$ as so:

{\begin{aligned}\sigma _{0}(24)&=\prod _{i=1}^{2}(a_{i}+1)\\&=(3+1)(1+1)=4\cdot 2=8.\end{aligned}}

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

Other properties and identities

For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and $\sigma_{0}(n)$ is even; for a square integer, one divisor (namely ${\sqrt {n}}$ ) is not paired with a distinct divisor and $\sigma_{0}(n)$ is odd. Similarly, the number $\sigma _{1}(n)$ is odd if and only if n is a square or twice a square.

We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, that is, 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, for example, $n = 2^k$ , then $\sigma (n)=2\cdot 2^{k}-1=2n-1,$ 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+1)(q+1) = n + 1 + (p+q), \,

\varphi(n) = (p-1)(q-1) = n + 1 - (p+q), \,

and

n + 1 = (\sigma(n) + \varphi(n))/2, \,

p + q = (\sigma(n) - \varphi(n))/2, \,

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

Then, the roots of:

(x-p)(x-q) = x^2 - (p+q)x + n = x^2 - [(\sigma(n) - \varphi(n))/2]x + [(\sigma(n) + \varphi(n))/2 - 1] = 0 \,

allow 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) - \varphi(n))/4 - \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}, \,

q = (\sigma(n) - \varphi(n))/4 + \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(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 the equality

\sigma_0(n) = \sigma_0(n + 1)

is true for an infinity of values of n, see A005237.

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)\quad\text{for}\quad s>1,s>a+1,

which for d(n) = σ₀(n) gives

\sum_{n=1}^\infty \frac{d(n)}{n^s} = \zeta^2(s)\quad\text{for}\quad s>1,

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.

Growth rate

In little-o notation, the divisor function satisfies the inequality (see page 296 of Apostol’s book^[4])

\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, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality (see Theorem 3.3 of Apostol’s book^[4])

\mbox{for all } x\geq1, \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}),

where $\gamma$ is Euler's gamma 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\leq 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 (Ramanujan 1997). The largest known value that violates the inequality is n=5040. In 1984 Guy Robin proved that the inequality is true for all n > 5040 if and only if the Riemann hypothesis is true (Robin 1984). This is Robin's theorem and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n > 5040 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).

Robin also proved, unconditionally, that the inequality

\ \sigma(n) < e^\gamma n \log \log n + \frac{0.6483\ n}{\log \log n}

holds for all n ≥ 3.

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

\sigma(n) < H_n + \ln(H_n)e^{H_n}

for every natural number n > 1, where $H_{n}$ is the nth harmonic number, (Lagarias 2002).

Notes

1 2 Long (1972, p. 46)
↑ Pettofrezzo & Byrkit (1970, p. 63)
↑ Pettofrezzo & Byrkit (1970, p. 58)
1 2 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, Zbl 0335.10001

References

Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis" (PDF), American Mathematical Monthly, 116 (3): 273–275, doi:10.4169/193009709X470128 .
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" (PDF), INTEGERS: the Electronic Journal of Combinatorial Number Theory, 11: A33
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, ISSN 1246-7405, MR 2394891, Zbl 1163.11059, arXiv:math.NT/0604314 , doi:10.5802/jtnb.591
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
Ivić, Aleksandar (1985), The Riemann zeta-function. The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, New York etc.: John Wiley & Sons, pp. 385–440, ISBN 0-471-80634-X, Zbl 0556.10026
Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, ISSN 0002-9890, JSTOR 2695443, MR 1908008, doi:10.2307/2695443
Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950
Ramanujan, Srinivasa (1997), "Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin", The Ramanujan Journal, 1 (2): 119–153, ISSN 1382-4090, MR 1606180, doi:10.1023/A:1009764017495
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77081766
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
Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, Zbl 1227.11002
Weisstein, Eric Wolfgang. "Divisor Function". MathWorld.

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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 Arithmetic number
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable Sociable Betrothed
Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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