# 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 σz(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

${\displaystyle \sigma _{z}(n)=\sum _{d\mid n}d^{z}\,\!,}$

where ${\displaystyle {d\mid n}}$ 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] (). When z 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) ().

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

{\displaystyle {\begin{aligned}\sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\\&=1+1+1+1+1+1=6,\end{aligned}}}

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

{\displaystyle {\begin{aligned}\sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\\&=1+2+3+4+6+12=28,\end{aligned}}}

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

{\displaystyle {\begin{aligned}s(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\\&=1+2+3+4+6=16.\end{aligned}}}

## Table of values

The cases x = 2 to 5 are listed in , x = 6 to 24 are listed in .

n factorization 𝜎0(n) 𝜎1(n) 𝜎2(n) 𝜎3(n) 𝜎4(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,

{\displaystyle {\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 pn# denotes the primorial,

${\displaystyle \sigma _{0}(p_{n}\#)=2^{n}}$

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

Clearly, ${\displaystyle 1<\sigma _{0}(n) for all ${\displaystyle n>2}$, and ${\displaystyle \sigma _{x}(n)>n}$ for all ${\displaystyle n>1}$, ${\displaystyle x>0}$ .

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

${\displaystyle \gcd(a,b)=1\Longrightarrow \sigma _{x}(ab)=\sigma _{x}(a)\sigma _{x}(b).}$

The consequence of this is that, if we write

${\displaystyle n=\prod _{i=1}^{r}p_{i}^{a_{i}}}$

where r = ω(n) is the number of distinct prime factors of n, pi is the ith prime factor, and ai is the maximum power of pi by which n is divisible, then we have: [4]

${\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}\sum _{j=0}^{a_{i}}p_{i}^{jx}=\prod _{i=1}^{r}\left(1+p_{i}^{x}+p_{i}^{2x}+\cdots +p_{i}^{a_{i}x}\right).}$

which, when x ≠ 0, is equivalent to the useful formula: [4]

${\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}{\frac {p_{i}^{(a_{i}+1)x}-1}{p_{i}^{x}-1}}.}$

When x = 0, d(n) is: [4]

${\displaystyle \sigma _{0}(n)=\prod _{i=1}^{r}(a_{i}+1).}$

This result can be directly deduced from the fact that all divisors of ${\displaystyle n}$ are uniquely determined by the distinct tuples ${\displaystyle (x_{1},x_{2},...,x_{i},...,x_{r})}$ of integers with ${\displaystyle 0\leq x_{i}\leq a_{i}}$ (i.e. ${\displaystyle a_{i}+1}$ independent choices for each ${\displaystyle x_{i}}$).

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 ${\displaystyle \sigma _{0}(24)}$ as so:

${\displaystyle \sigma _{0}(24)=\prod _{i=1}^{2}(a_{i}+1)=(3+1)(1+1)=4\cdot 2=8.}$

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:[5][6][7]

{\displaystyle {\begin{aligned}\sigma (n)&=\sigma (n-1)+\sigma (n-2)-\sigma (n-5)-\sigma (n-7)+\sigma (n-12)+\sigma (n-15)+\cdots \\[12mu]&=\sum _{i\in \mathbb {N} }(-1)^{i+1}\left(\sigma \left(n-{\frac {1}{2}}\left(3i^{2}-i\right)\right)+\sigma \left(n-{\frac {1}{2}}\left(3i^{2}+i\right)\right)\right)\end{aligned}}}

where ${\displaystyle \sigma (0)=n}$ if it occurs and ${\displaystyle \sigma (x)=0}$ for ${\displaystyle x<0}$, and ${\displaystyle {\tfrac {1}{2}}\left(3i^{2}\mp i\right)}$ are consecutive pairs of generalized pentagonal numbers (, 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 ${\displaystyle \sigma _{0}(n)}$ is even; for a square integer, one divisor (namely ${\displaystyle {\sqrt {n}}}$) is not paired with a distinct divisor and ${\displaystyle \sigma _{0}(n)}$ is odd. Similarly, the number ${\displaystyle \sigma _{1}(n)}$ is odd if and only if n is a square or twice a square.[8]

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 n is a power of 2, ${\displaystyle n=2^{k}}$, then ${\displaystyle \sigma (n)=2\cdot 2^{k}-1=2n-1}$ and ${\displaystyle s(n)=n-1}$, which makes n almost-perfect.

As an example, for two primes ${\displaystyle p,q:p, let

${\displaystyle n=p\,q}$.

Then

${\displaystyle \sigma (n)=(p+1)(q+1)=n+1+(p+q),}$
${\displaystyle \varphi (n)=(p-1)(q-1)=n+1-(p+q),}$

and

${\displaystyle n+1=(\sigma (n)+\varphi (n))/2,}$
${\displaystyle p+q=(\sigma (n)-\varphi (n))/2,}$

where ${\displaystyle \varphi (n)}$ is Euler's totient function.

Then, the roots of

${\displaystyle (x-p)(x-q)=x^{2}-(p+q)x+n=x^{2}-[(\sigma (n)-\varphi (n))/2]x+[(\sigma (n)+\varphi (n))/2-1]=0}$

express p and q in terms of σ(n) and φ(n) only, requiring no knowledge of n or ${\displaystyle p+q}$, as

${\displaystyle p=(\sigma (n)-\varphi (n))/4-{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}},}$
${\displaystyle q=(\sigma (n)-\varphi (n))/4+{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}}.}$

Also, knowing n and either ${\displaystyle \sigma (n)}$ or ${\displaystyle \varphi (n)}$, or, alternatively, ${\displaystyle p+q}$ and either ${\displaystyle \sigma (n)}$ or ${\displaystyle \varphi (n)}$ allows an easy recovery of p and q.

In 1984, Roger Heath-Brown proved that the equality

${\displaystyle \sigma _{0}(n)=\sigma _{0}(n+1)}$

is true for infinitely many values of n, see .

## Series relations

Two Dirichlet series involving the divisor function are: [9]

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a)\quad {\text{for}}\quad s>1,s>a+1,}$

which for d(n) = σ0(n) gives: [9]

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

and a Ramanujan identity[10]

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}},}$

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

A Lambert series involving the divisor function is: [11]

${\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }n^{a}q^{j\,n}=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}}$

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.

For ${\displaystyle k>0}$, there is an explicit series representation with Ramanujan sums ${\displaystyle c_{m}(n)}$ as :[12]

${\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\sum _{m=1}^{\infty }{\frac {c_{m}(n)}{m^{k+1}}}.}$

The computation of the first terms of ${\displaystyle c_{m}(n)}$ shows its oscillations around the "average value" ${\displaystyle \zeta (k+1)n^{k}}$:

${\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\left[1+{\frac {(-1)^{n}}{2^{k+1}}}+{\frac {2\cos {\frac {2\pi n}{3}}}{3^{k+1}}}+{\frac {2\cos {\frac {\pi n}{2}}}{4^{k+1}}}+\cdots \right]}$

## Growth rate

In little-o notation, the divisor function satisfies the inequality:[13][14]

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

More precisely, Severin Wigert showed that:[14]

${\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)}{\log n/\log \log n}}=\log 2.}$

On the other hand, since there are infinitely many prime numbers,[14]

${\displaystyle \liminf _{n\to \infty }d(n)=2.}$

In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:[15][16]

${\displaystyle {\mbox{for all }}x\geq 1,\sum _{n\leq x}d(n)=x\log x+(2\gamma -1)x+O({\sqrt {x}}),}$

where ${\displaystyle \gamma }$ is Euler's gamma constant. Improving the bound ${\displaystyle O({\sqrt {x}})}$ 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: [17]

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

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:

${\displaystyle \lim _{n\to \infty }{\frac {1}{\log n}}\prod _{p\leq n}{\frac {p}{p-1}}=e^{\gamma },}$

where p denotes a prime.

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

${\displaystyle \ \sigma (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:

${\displaystyle \ \sigma (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:

${\displaystyle \sigma (n)

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

## Notes

1. ^ a b Long (1972, p. 46)
2. ^ Pettofrezzo & Byrkit (1970, p. 63)
3. ^ Pettofrezzo & Byrkit (1970, p. 58)
4. ^ a b c Hardy & Wright (2008), pp. 310 f, §16.7.
5. ^ Euler, Leonhard; Bell, Jordan (2004). "An observation on the sums of divisors". arXiv:math/0411587.
6. ^ https://scholarlycommons.pacific.edu/euler-works/175/, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs
7. ^ https://scholarlycommons.pacific.edu/euler-works/542/, De mirabilis proprietatibus numerorum pentagonalium
8. ^
9. ^ a b Hardy & Wright (2008), pp. 326–328, §17.5.
10. ^ Hardy & Wright (2008), pp. 334–337, §17.8.
11. ^ Hardy & Wright (2008), pp. 338–341, §17.10.
12. ^ E. Krätzel (1981). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. p. 130. (German)
13. ^ Apostol (1976), p. 296.
14. ^ a b c Hardy & Wright (2008), pp. 342–347, §18.1.
15. ^ Apostol (1976), Theorem 3.3.
16. ^ Hardy & Wright (2008), pp. 347–350, §18.2.
17. ^ Hardy & Wright (2008), pp. 469–471, §22.9.