Divisor function

File:Divisor.svg

Divisor function σ₀(n) up to n = 250

File:Sigma function.svg

Sigma function σ₁(n) up to n = 250

File:Divisor square.svg

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

File:Divisor cube.svg

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_z(n), for a real or complex number z, is defined as the sum of the zth powers of the positive divisors of n. It can be expressed in sigma notation as

<math>\sigma_z(n)=\sum_{d\mid n} d^z\,\! ,</math>

where <math>{d\mid n}</math> 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<ref name="Long 1972 46">Template:Harvtxt</ref><ref>Template:Harvtxt</ref> (Template:OEIS2C). When z is 1, the function is called the sigma function or sum-of-divisors function,<ref name="Long 1972 46"/><ref>Template:Harvtxt</ref> and the subscript is often omitted, so σ(n) is the same as σ₁(n) (Template:OEIS2C).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, Template:OEIS2C), 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:

<math>

\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} </math>

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

<math>

\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} </math>

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

<math>

\begin{align} s(12) & = 1^1 + 2^1 + 3^1 + 4^1 + 6^1 \\ & = 1 + 2 + 3 + 4 + 6 = 16. \end{align} </math>

σ₋₁(n) is sometimes called the abundancy index of n, and we have:

<math>

\begin{align} \sigma_{-1}(12) & = 1^{-1} + 2^{-1} + 3^{-1} + 4^{-1} + 6^{-1} + 12^{-1} \\[6pt] & = \tfrac11 + \tfrac12 + \tfrac13 + \tfrac14 + \tfrac16 + \tfrac1{12} \\[6pt] & = \tfrac{12}{12} + \tfrac6{12} + \tfrac4{12} + \tfrac3{12} + \tfrac2{12} + \tfrac1{12} \\[6pt] & = \tfrac{12 + 6 + 4 + 3 + 2 + 1}{12} = \tfrac{28}{12} = \tfrac73 = \tfrac{\sigma_1(12)}{12} \end{align} </math>

Table of values

The cases x = 2 to 5 are listed in Template:OEIS2C through Template:OEIS2C, x = 6 to 24 are listed in Template:OEIS2C through Template:OEIS2C.

n	prime factorization	Template:Sigma0(n)	Template:Sigma1(n)	Template:Sigma2(n)	Template:Sigma3(n)	Template:Sigma4(n)
1	1	1	1	1	1	1
2	2	2	3	5	9	17
3	3	2	4	10	28	82
4	22	3	7	21	73	273
5	5	2	6	26	126	626
6	2×3	4	12	50	252	1394
7	7	2	8	50	344	2402
8	23	4	15	85	585	4369
9	32	3	13	91	757	6643
10	2×5	4	18	130	1134	10642
11	11	2	12	122	1332	14642
12	22×3	6	28	210	2044	22386
13	13	2	14	170	2198	28562
14	2×7	4	24	250	3096	40834
15	3×5	4	24	260	3528	51332
16	24	5	31	341	4681	69905
17	17	2	18	290	4914	83522
18	2×32	6	39	455	6813	112931
19	19	2	20	362	6860	130322
20	22×5	6	42	546	9198	170898
21	3×7	4	32	500	9632	196964
22	2×11	4	36	610	11988	248914
23	23	2	24	530	12168	279842
24	23×3	8	60	850	16380	358258
25	52	3	31	651	15751	391251
26	2×13	4	42	850	19782	485554
27	33	4	40	820	20440	538084
28	22×7	6	56	1050	25112	655746
29	29	2	30	842	24390	707282
30	2×3×5	8	72	1300	31752	872644
31	31	2	32	962	29792	923522
32	25	6	63	1365	37449	1118481
33	3×11	4	48	1220	37296	1200644
34	2×17	4	54	1450	44226	1419874
35	5×7	4	48	1300	43344	1503652
36	22×32	9	91	1911	55261	1813539
37	37	2	38	1370	50654	1874162
38	2×19	4	60	1810	61740	2215474
39	3×13	4	56	1700	61544	2342084
40	23×5	8	90	2210	73710	2734994
41	41	2	42	1682	68922	2825762
42	2×3×7	8	96	2500	86688	3348388
43	43	2	44	1850	79508	3418802
44	22×11	6	84	2562	97236	3997266
45	32×5	6	78	2366	95382	4158518
46	2×23	4	72	2650	109512	4757314
47	47	2	48	2210	103824	4879682
48	24×3	10	124	3410	131068	5732210
49	72	3	57	2451	117993	5767203
50	2×52	6	93	3255	141759	6651267

Properties

Formulas at prime powers

For a prime number p,

<math>\begin{align}

\sigma_0(p) & = 2 \\ \sigma_0(p^n) & = n+1 \\ \sigma_1(p) & = p+1 \end{align}</math>

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

<math> \sigma_0(p_n\#) = 2^n </math>

since n prime factors allow a sequence of binary selection (<math>p_{i}</math> or 1) from n terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a power of two; instead, the smallest such number may be obtained by multiplying together the first n Fermi–Dirac primes, prime powers whose exponent is a power of two.<ref>Template:Citation; see section 47, pp. 405–406, reproduced in Collected Papers of Srinivasa Ramanujan, Cambridge Univ. Press, 2015, pp. 124–125</ref>

Clearly, <math>1 < \sigma_0(n) < n</math> for all <math>n > 2</math>, and <math>\sigma_x(n) > n </math> for all <math>n > 1</math>, <math>x > 0</math> .

The divisor function is multiplicative (since each divisor c of the product mn with <math>\gcd(m, n) = 1</math> distinctively correspond to a divisor a of m and a divisor b of n), but not completely multiplicative:

<math>\gcd(a, b)=1 \Longrightarrow \sigma_x(ab)=\sigma_x(a)\sigma_x(b).</math>

The consequence of this is that, if we write

<math>n = \prod_{i=1}^r p_i^{a_i}</math>

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: Template:Sfnp

<math>\sigma_x(n) = \prod_{i=1}^r \sum_{j=0}^{a_i} p_i^{j x} = \prod_{i=1}^r \left (1 + p_i^x + p_i^{2x} + \cdots + p_i^{a_i x} \right ).</math>

which, when x ≠ 0, is equivalent to the useful formula: Template:Sfnp

<math>\sigma_x(n) = \prod_{i=1}^{r} \frac{p_{i}^{(a_{i}+1)x}-1}{p_{i}^x-1}.</math>

When x = 0, <math>\sigma_0(n)</math> is: Template:Sfnp

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

This result can be directly deduced from the fact that all divisors of <math>n</math> are uniquely determined by the distinct tuples <math>(x_1, x_2, ..., x_i, ..., x_r)</math> of integers with <math>0 \le x_i \le a_i</math> (i.e. <math>a_i+1</math> independent choices for each <math>x_i</math>).

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 <math>\sigma_0(24)</math> as so:

<math>\sigma_0(24) = \prod_{i=1}^{2} (a_i+1) = (3 + 1)(1 + 1) = 4 \cdot 2 = 8.</math>

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

Other properties and identities

Euler proved the remarkable recurrence:<ref>Template:Cite arXiv</ref><ref>https://scholarlycommons.pacific.edu/euler-works/175/, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs</ref><ref>https://scholarlycommons.pacific.edu/euler-works/542/, De mirabilis proprietatibus numerorum pentagonalium</ref>

<math>\begin{align}

\sigma_1(n) &= \sigma_1(n-1)+\sigma_1(n-2)-\sigma_1(n-5)-\sigma_1(n-7)+\sigma_1(n-12)+\sigma_1(n-15)+ \cdots \\[12mu]

&= \sum_{i\in\N} (-1)^{i+1}\left( \sigma_1 \left( n-\frac{1}{2} \left( 3i^2-i \right) \right) + \sigma_1 \left( n-\frac{1}{2} \left( 3i^2+i \right) \right) \right),

\end{align}</math>

where <math>\sigma_1(0)=n</math> if it occurs and <math>\sigma_1(x)=0</math> for <math>x < 0</math>, and <math>\tfrac{1}{2} \left( 3i^2 \mp i \right)</math> are consecutive pairs of generalized pentagonal numbers (Template:OEIS2C, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his pentagonal number theorem.

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

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 used to recognize perfect numbers, which are the n such that 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 Template:Mvar is a power of 2, <math>n = 2^k</math>, then <math>\sigma(n) = 2 \cdot 2^k - 1 = 2n - 1</math> and <math>s(n) = n - 1</math>, which makes n almost-perfect.

As an example, for two primes <math>p,q:p<q</math>, let

<math>n = p\,q</math>.

Then

<math>\sigma(n) = (p+1)(q+1) = n + 1 + (p+q), </math>

<math>\varphi(n) = (p-1)(q-1) = n + 1 - (p+q), </math>

and

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

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

where <math>\varphi(n)</math> is Euler's totient function.

Then, the roots of

<math>(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 </math>

express p and q in terms of σ(n) and φ(n) only, requiring no knowledge of n or <math>p+q</math>, as

<math>p = (\sigma(n) - \varphi(n))/4 - \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}, </math>

<math>q = (\sigma(n) - \varphi(n))/4 + \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}. </math>

Also, knowing Template:Mvar and either <math>\sigma(n)</math> or <math>\varphi(n)</math>, or, alternatively, <math>p+q</math> and either <math>\sigma(n)</math> or <math>\varphi(n)</math> allows an easy recovery of p and q.

In 1984, Roger Heath-Brown proved that the equality

<math>\sigma_0(n) = \sigma_0(n + 1)</math>

is true for infinitely many values of Template:Mvar, see Template:OEIS2C.

Dirichlet convolutions

Template:Main article By definition:<math display="block">\sigma = \operatorname{Id} * \mathbf 1</math>By Möbius inversion:<math display="block">\operatorname{Id} = \sigma * \mu </math>

Series relations

Two Dirichlet series involving the divisor function are: Template:Sfnp

<math>\sum_{n=1}^\infty \frac{\sigma_{a}(n)}{n^s} = \zeta(s) \zeta(s-a)\quad\text{for}\quad s>1,s>a+1,</math>

where <math>\zeta</math> is the Riemann zeta function. The series for d(n) = σ₀(n) gives: Template:Sfnp

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

and a Ramanujan identityTemplate:Sfnp

<math>\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)},</math>

which is a special case of the Rankin–Selberg convolution.

A Lambert series involving the divisor function is: Template:Sfnp

<math>\sum_{n=1}^\infty q^n \sigma_a(n) = \sum_{n=1}^\infty \sum_{j=1}^\infty n^a q^{j\,n} = \sum_{n=1}^\infty \frac{n^a q^n}{1-q^n} = \sum_{n=1}^\infty \operatorname{Li}_{-a}(q^n)</math>

for arbitrary complex |q| ≤ 1 and a (<math>\operatorname{Li}</math> is the polylogarithm). This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.

For <math>k>0</math>, there is an explicit series representation with Ramanujan sums <math> c_m(n) </math> as :<ref>Template:Cite book (German)</ref>

<math>\sigma_k(n) = \zeta(k+1)n^k\sum_{m=1}^\infty \frac {c_m(n)}{m^{k+1}}.</math>

The computation of the first terms of <math>c_m(n)</math> shows its oscillations around the "average value" <math>\zeta(k+1)n^k</math>:

<math>\sigma_k(n) = \zeta(k+1)n^k \left[ 1 + \frac{(-1)^n}{2^{k+1}} + \frac{2\cos\frac {2\pi n}{3}}{3^{k+1}} + \frac{2\cos\frac {\pi n}{2}}{4^{k+1}} + \cdots\right]</math>

Growth rate

In little-o notation, the divisor function satisfies the inequality:Template:SfnpTemplate:Sfnp

<math>\mbox{for all }\varepsilon>0,\quad d(n)=o(n^\varepsilon).</math>

More precisely, Severin Wigert showed that:Template:Sfnp

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

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

<math>\liminf_{n\to\infty} d(n)=2.</math>

In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:Template:SfnpTemplate:Sfnp

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

where <math>\gamma</math> is Euler's gamma constant. Improving the bound <math>O(\sqrt{x})</math> 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: Template:Sfnp

<math>

\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma, </math>

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 Template:Harv. His proof uses Mertens' third theorem, which says that:

<math>\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^\gamma,</math>

where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, Robin's inequality

<math>\ \sigma(n) < e^\gamma n \log \log n </math> (where γ is the Euler–Mascheroni constant)

holds for all sufficiently large n Template:Harv. 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 Template:Harv. 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 Template:Harv. 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 Template:Harv.

Robin also proved, unconditionally, that the inequality:

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

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:

<math> \sigma(n) < H_n + e^{H_n}\log(H_n)</math>

for every natural number n > 1, where <math>H_n</math> is the nth harmonic number, Template:Harv.

See also

• Divisor sum convolutions, lists a few identities involving the divisor functions
• Euler's totient function, Euler's phi function
• Refactorable number
• Table of divisors
• Unitary divisor

References

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• Template:Apostol IANT
• Bach, Eric; Shallit, Jeffrey, Algorithmic Number Theory, volume 1, 1996, MIT Press. Template:ISBN, see page 234 in section 8.8.
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External links

• Template:Mathworld
• Template: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.