divisor function

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

:$\backslash sigma\_z(n)=\backslash sum\_\{d\backslash mid\; n\}\; d^z\backslash ,\backslash !\; ,$

where $\{d\backslash mid\; n\}$ 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{{harvtxt|Long|1972|p=46}}{{harvtxt|Pettofrezzo|Byrkit|1970|p=63}} ({{OEIS2C|id=A000005}}). When z is 1, the function is called the sigma function or sum-of-divisors function,{{harvtxt|Pettofrezzo|Byrkit|1970|p=58}} and the subscript is often omitted, so σ(n) is the same as σ₁(n) ({{OEIS2C|id=A000203}}).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, {{OEIS2C|id=A001065}}), and equals σ₁(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

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}

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

: $$

\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}

The cases x = 2 to 5 are listed in {{OEIS2C|A001157}} through {{OEIS2C|A001160}}, x = 6 to 24 are listed in {{OEIS2C|A013954}} through {{OEIS2C|A013972}}.

{{clear}}

For a prime number p,

:$\backslash begin\{align\}$

\sigma_0(p) & = 2 \\

\sigma_0(p^n) & = n+1 \\

\sigma_1(p) & = p+1

\end{align}

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

:$\backslash sigma\_0(p\_n\backslash \#)\; =\; 2^n$

since n prime factors allow a sequence of binary selection ($p\_\{i\}$ 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.{{citation

| last = Ramanujan | first = S. | author-link = Srinivasa Ramanujan

| doi = 10.1112/plms/s2_14.1.347

| issue = 1

| journal = Proceedings of the London Mathematical Society

| pages = 347–409

| title = Highly Composite Numbers

| volume = s2-14

| year = 1915| url = https://zenodo.org/record/1433496 }}; see section 47, pp. 405–406, reproduced in Collected Papers of Srinivasa Ramanujan, Cambridge Univ. Press, 2015, [https://books.google.com/books?id=h1G2CgAAQBAJ&pg=PA124 pp. 124–125]

Clearly, for all $n\; >\; 2$, and $\backslash sigma\_x(n)\; >\; n$ for all $n\; >\; 1$, $x\; >\; 0$ .

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

:$\backslash gcd(a,\; b)=1\; \backslash Longrightarrow\; \backslash sigma\_x(ab)=\backslash sigma\_x(a)\backslash sigma\_x(b).$

The consequence of this is that, if we write

:$n\; =\; \backslash prod\_\{i=1\}^r\; p\_i^\{a\_i\}$

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: {{sfnp|Hardy|Wright|2008|pp=310 f|loc=§16.7}}

:$\backslash sigma\_x(n)\; =\; \backslash prod\_\{i=1\}^r\; \backslash sum\_\{j=0\}^\{a\_i\}\; p\_i^\{j\; x\}\; =\; \backslash prod\_\{i=1\}^r\; \backslash left\; (1\; +\; p\_i^x\; +\; p\_i^\{2x\}\; +\; \backslash cdots\; +\; p\_i^\{a\_i\; x\}\; \backslash right\; ).$

which, when x ≠ 0, is equivalent to the useful formula: {{sfnp|Hardy|Wright|2008|pp=310 f|loc=§16.7}}

:$\backslash sigma\_x(n)\; =\; \backslash prod\_\{i=1\}^\{r\}\; \backslash frac\{p\_\{i\}^\{(a\_\{i\}+1)x\}-1\}\{p\_\{i\}^x-1\}.$

When x = 0, $\backslash sigma\_0(n)$ is: {{sfnp|Hardy|Wright|2008|pp=310 f|loc=§16.7}}

:$\backslash sigma\_0(n)=\backslash prod\_\{i=1\}^r\; (a\_i+1).$

This result can be directly deduced from the fact that all divisors of $n$ are uniquely determined by the distinct tuples $(x\_1,\; x\_2,\; ...,\; x\_i,\; ...,\; x\_r)$ of integers with $0\; \backslash le\; x\_i\; \backslash le\; a\_i$ (i.e. $a\_i+1$ independent choices for each $x\_i$).

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 $\backslash sigma\_0(24)$ as so:

: $\backslash sigma\_0(24)\; =\; \backslash prod\_\{i=1\}^\{2\}\; (a\_i+1)\; =\; (3\; +\; 1)(1\; +\; 1)\; =\; 4\; \backslash cdot\; 2\; =\; 8.$

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

Euler proved the remarkable recurrence:{{Cite arXiv |eprint = math/0411587|last1 = Euler|first1 = Leonhard|title = An observation on the sums of divisors|last2 = Bell|first2 = Jordan|year = 2004}}https://scholarlycommons.pacific.edu/euler-works/175/, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurshttps://scholarlycommons.pacific.edu/euler-works/542/, De mirabilis proprietatibus numerorum pentagonalium

:$\backslash begin\{align\}$

\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}

where $\backslash sigma\_1(0)=n$ if it occurs and $\backslash sigma\_1(x)=0$ for , and $\backslash tfrac\{1\}\{2\}\; \backslash left(\; 3i^2\; \backslash mp\; i\; \backslash right)$ are consecutive pairs of generalized pentagonal numbers ({{OEIS2C|A001318}}, 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 $\backslash sigma\_\{0\}(n)$ is even; for a square integer, one divisor (namely $\backslash sqrt\; n$) is not paired with a distinct divisor and $\backslash sigma\_\{0\}(n)$ is odd. Similarly, the number $\backslash sigma\_\{1\}(n)$ is odd if and only if n is a square or twice a square.{{sfnp|Gioia|Vaidya|1967}}

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

As an example, for two primes

:$n\; =\; p\backslash ,q$.

Then

:$\backslash sigma(n)\; =\; (p+1)(q+1)\; =\; n\; +\; 1\; +\; (p+q),$

:$\backslash varphi(n)\; =\; (p-1)(q-1)\; =\; n\; +\; 1\; -\; (p+q),$

and

:$n\; +\; 1\; =\; (\backslash sigma(n)\; +\; \backslash varphi(n))/2,$

:$p\; +\; q\; =\; (\backslash sigma(n)\; -\; \backslash varphi(n))/2,$

where $\backslash varphi(n)$ is Euler's totient function.

Then, the roots of

:$(x-p)(x-q)\; =\; x^2\; -\; (p+q)x\; +\; n\; =\; x^2\; -\; [(\backslash sigma(n)\; -\; \backslash varphi(n))/2]x\; +\; [(\backslash sigma(n)\; +\; \backslash varphi(n))/2\; -\; 1]\; =\; 0$

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

:$p\; =\; (\backslash sigma(n)\; -\; \backslash varphi(n))/4\; -\; \backslash sqrt\{[(\backslash sigma(n)\; -\; \backslash varphi(n))/4]^2\; -\; [(\backslash sigma(n)\; +\; \backslash varphi(n))/2\; -\; 1]\},$

:$q\; =\; (\backslash sigma(n)\; -\; \backslash varphi(n))/4\; +\; \backslash sqrt\{[(\backslash sigma(n)\; -\; \backslash varphi(n))/4]^2\; -\; [(\backslash sigma(n)\; +\; \backslash varphi(n))/2\; -\; 1]\}.$

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

In 1984, Roger Heath-Brown proved that the equality

:$\backslash sigma\_0(n)\; =\; \backslash sigma\_0(n\; +\; 1)$

is true for infinitely many values of {{mvar|n}}, see {{OEIS2C|A005237}}.

{{Main article|Dirichlet convolution}}

By definition:$$\backslash sigma\; =\; \backslash operatorname\{Id\}\; *\; \backslash mathbf\; 1$$By Möbius inversion:$$\backslash operatorname\{Id\}\; =\; \backslash sigma\; *\; \backslash mu$$

Two Dirichlet series involving the divisor function are: {{sfnp|Hardy|Wright|2008|pp=326-328|loc=§17.5}}

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

where $\backslash zeta$ is the Riemann zeta function. The series for d(n) = σ₀(n) gives: {{sfnp|Hardy|Wright|2008|pp=326-328|loc=§17.5}}

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

and a Ramanujan identity{{sfnp|Hardy|Wright|2008|pp=334-337|loc=§17.8}}

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash sigma\_a(n)\backslash sigma\_b(n)\}\{n^s\}\; =\; \backslash frac\{\backslash zeta(s)\; \backslash zeta(s-a)\; \backslash zeta(s-b)\; \backslash zeta(s-a-b)\}\{\backslash zeta(2s-a-b)\},$

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

A Lambert series involving the divisor function is: {{sfnp|Hardy|Wright|2008|pp=338-341|loc=§17.10}}

:$\backslash sum\_\{n=1\}^\backslash infty\; q^n\; \backslash sigma\_a(n)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash sum\_\{j=1\}^\backslash infty\; n^a\; q^\{j\backslash ,n\}\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{n^a\; q^n\}\{1-q^n\}\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash operatorname\{Li\}\_\{-a\}(q^n)$

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

For $k>0$, there is an explicit series representation with Ramanujan sums $c\_m(n)$ as :{{cite book |author=E. Krätzel |title=Zahlentheorie |publisher=VEB Deutscher Verlag der Wissenschaften |place =Berlin |year=1981 |pages=130}} (German)

:$\backslash sigma\_k(n)\; =\; \backslash zeta(k+1)n^k\backslash sum\_\{m=1\}^\backslash infty\; \backslash frac\; \{c\_m(n)\}\{m^\{k+1\}\}.$

The computation of the first terms of $c\_m(n)$ shows its oscillations around the "average value" $\backslash zeta(k+1)n^k$:

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

In little-o notation, the divisor function satisfies the inequality:{{sfnp|Apostol|1976|p=296}}{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}

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

More precisely, Severin Wigert showed that:{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}

:$\backslash limsup\_\{n\backslash to\backslash infty\}\backslash frac\{\backslash log\; d(n)\}\{\backslash log\; n/\backslash log\backslash log\; n\}=\backslash log2.$

On the other hand, since there are infinitely many prime numbers,{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}

:$\backslash liminf\_\{n\backslash to\backslash infty\}\; d(n)=2.$

In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:{{sfnp|Apostol|1976|loc=Theorem 3.3}}{{sfnp|Hardy|Wright|2008|pp=347-350|loc=§18.2}}

:$\backslash mbox\{for\; all\; \}\; x\backslash geq1,\; \backslash sum\_\{n\backslash leq\; x\}d(n)=x\backslash log\; x+(2\backslash gamma-1)x+O(\backslash sqrt\{x\}),$

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

{{anchor|Robin's theorem|Robin's inequality|Grönwall's theorem}}

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: {{sfnp|Hardy|Wright|2008|pp=469-471|loc=§22.9}}

:$$

\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 {{harv|Grönwall|1913}}. His proof uses Mertens' third theorem, which says that:

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{1\}\{\backslash log\; n\}\backslash prod\_\{p\backslash le\; n\}\backslash frac\{p\}\{p-1\}=e^\backslash gamma,$

where p denotes a prime.

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

: (where γ is the Euler–Mascheroni constant)

holds for all sufficiently large n {{harv|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 {{harv|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 {{harv|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 {{Harv|Choie|Lichiardopol|Moree|Solé|2007}}.

Robin also proved, unconditionally, that the inequality:

:

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:

:

for every natural number n > 1, where $H\_n$ is the nth harmonic number, {{harv|Lagarias|2002}}.

• 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

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• {{mathworld|urlname=DivisorFunction|title=Divisor Function}}
• {{mathworld|urlname=RobinsTheorem|title=Robin's Theorem}}
• [http://mathstat.carleton.ca/~williams/papers/pdf/249.pdf 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.

{{Divisor classes}}

{{DEFAULTSORT:Divisor function}}

Category:Analytic number theory

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