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{{redirect|Robin's theorem|Robbins' theorem in graph theory|Robbins' theorem}}
{{short description|Arithmetic function related to the divisors of an integer}}

[[Image:Divisor.svg|thumb|right|Divisor function ''σ''<sub>0</sub>(''n'') up to ''n''&nbsp;=&nbsp;250]]
[[Image:Sigma function.svg|thumb|right|Sigma function ''σ''<sub>1</sub>(''n'') up to ''n''&nbsp;=&nbsp;250]]
[[Image:Divisor square.svg|thumb|right|Sum of the squares of divisors, ''σ''<sub>2</sub>(''n''), up to ''n''&nbsp;=&nbsp;250]]
[[Image:Divisor cube.svg|thumb|right|Sum of cubes of divisors, ''σ''<sub>3</sub>(''n'') up to ''n''&nbsp;=&nbsp;250]]
In [[mathematics]], and specifically in [[number theory]], a '''divisor function''' is an [[arithmetic function]] related to the [[divisor]]s 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 form]]s. Divisor functions were studied by [[Ramanujan]], who gave a number of important [[Modular arithmetic|congruences]] and [[identity (mathematics)|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''<sub>''z''</sub>(''n''), for a real or complex number ''z'', is defined as the [[summation|sum]] of the ''z''th [[Exponentiation|powers]] of the positive [[divisor]]s of ''n''.  It can be expressed in [[Summation#Capital-sigma notation|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 ''σ''<sub>0</sub>(''n''), or the '''number-of-divisors function'''<ref name="Long 1972 46">{{harvtxt|Long|1972|p=46}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=63}}</ref> ({{OEIS2C|id=A000005}}). When ''z'' is 1, the function is called the '''sigma function''' or '''sum-of-divisors function''',<ref name="Long 1972 46"/><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=58}}</ref> and the subscript is often omitted, so ''σ''(''n'') is the same as ''σ''<sub>1</sub>(''n'') ({{OEIS2C|id=A000203}}).

The '''[[aliquot sum]]''' ''s''(''n'') of ''n'' is the sum of the [[proper divisor]]s (that is, the divisors excluding ''n'' itself, {{OEIS2C|id=A001065}}), and equals ''σ''<sub>1</sub>(''n'')&nbsp;&minus;&nbsp;''n''; the [[aliquot sequence]] of ''n'' is formed by repeatedly applying the aliquot sum function.

==Example==
For example, ''σ''<sub>0</sub>(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 ''σ''<sub>1</sub>(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''<sub>−1</sub>(''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 {{OEIS2C|A001157}} through {{OEIS2C|A001160}}, ''x'' = 6 to 24 are listed in {{OEIS2C|A013954}} through {{OEIS2C|A013972}}.

{| class="wikitable" style="text-align:right; float:left"
! ''n'' !! prime factorization !! {{sigma}}<sub>0</sub>(''n'')!!  {{sigma}}<sub>1</sub>(''n'')!!  {{sigma}}<sub>2</sub>(''n'')!!  {{sigma}}<sub>3</sub>(''n'')!!  {{sigma}}<sub>4</sub>(''n'')
|-
| 1||style='text-align:center;'| 1|| 1|| 1|| 1|| 1|| 1
|-style="background-color:#ddeeff;"
| 2||style='text-align:center;'| 2|| 2|| 3|| 5|| 9|| 17
|-style="background-color:#ddeeff;"
| 3||style='text-align:center;'| 3|| 2|| 4|| 10|| 28|| 82
|-
| 4||style='text-align:center;'| 2<sup>2</sup>|| 3|| 7|| 21|| 73|| 273
|-style="background-color:#ddeeff;"
| 5||style='text-align:center;'| 5|| 2|| 6|| 26|| 126|| 626
|-
| 6||style='text-align:center;'| 2×3|| 4|| 12|| 50|| 252|| 1394
|-style="background-color:#ddeeff;"
| 7||style='text-align:center;'| 7|| 2|| 8|| 50|| 344|| 2402
|-
| 8||style='text-align:center;'| 2<sup>3</sup>|| 4|| 15|| 85|| 585|| 4369
|-
| 9||style='text-align:center;'| 3<sup>2</sup>|| 3|| 13|| 91|| 757|| 6643
|-
| 10||style='text-align:center;'| 2×5|| 4|| 18|| 130|| 1134|| 10642
|-style="background-color:#ddeeff;"
| 11||style='text-align:center;'| 11|| 2|| 12|| 122|| 1332|| 14642
|-
| 12||style='text-align:center;'| 2<sup>2</sup>×3|| 6|| 28|| 210|| 2044|| 22386
|-style="background-color:#ddeeff;"
| 13||style='text-align:center;'| 13|| 2|| 14|| 170|| 2198|| 28562
|-
| 14||style='text-align:center;'| 2×7|| 4|| 24|| 250|| 3096|| 40834
|-
| 15||style='text-align:center;'| 3×5|| 4|| 24|| 260|| 3528|| 51332
|-
| 16||style='text-align:center;'| 2<sup>4</sup>|| 5|| 31|| 341|| 4681|| 69905
|-style="background-color:#ddeeff;"
| 17||style='text-align:center;'| 17|| 2|| 18|| 290|| 4914|| 83522
|-
| 18||style='text-align:center;'| 2×3<sup>2</sup>|| 6|| 39|| 455|| 6813|| 112931
|-style="background-color:#ddeeff;"
| 19||style='text-align:center;'| 19|| 2|| 20|| 362|| 6860|| 130322
|-
| 20||style='text-align:center;'| 2<sup>2</sup>×5|| 6|| 42|| 546|| 9198|| 170898
|-
| 21||style='text-align:center;'| 3×7|| 4|| 32|| 500|| 9632|| 196964
|-
| 22||style='text-align:center;'| 2×11|| 4|| 36|| 610|| 11988|| 248914
|-style="background-color:#ddeeff;"
| 23||style='text-align:center;'| 23|| 2|| 24|| 530|| 12168|| 279842
|-
| 24||style='text-align:center;'| 2<sup>3</sup>×3|| 8|| 60|| 850|| 16380|| 358258
|-
| 25||style='text-align:center;'| 5<sup>2</sup>|| 3|| 31|| 651|| 15751|| 391251
|-
| 26||style='text-align:center;'| 2×13|| 4|| 42|| 850|| 19782|| 485554
|-
| 27||style='text-align:center;'| 3<sup>3</sup>|| 4|| 40|| 820|| 20440|| 538084
|-
| 28||style='text-align:center;'| 2<sup>2</sup>×7|| 6|| 56|| 1050|| 25112|| 655746
|-style="background-color:#ddeeff;"
| 29||style='text-align:center;'| 29|| 2|| 30|| 842|| 24390|| 707282
|-
| 30||style='text-align:center;'| 2×3×5|| 8|| 72|| 1300|| 31752|| 872644
|-style="background-color:#ddeeff;"
| 31||style='text-align:center;'| 31|| 2|| 32|| 962|| 29792|| 923522
|-
| 32||style='text-align:center;'| 2<sup>5</sup>|| 6|| 63|| 1365|| 37449|| 1118481
|-
| 33||style='text-align:center;'| 3×11|| 4|| 48|| 1220|| 37296|| 1200644
|-
| 34||style='text-align:center;'| 2×17|| 4|| 54|| 1450|| 44226|| 1419874
|-
| 35||style='text-align:center;'| 5×7|| 4|| 48|| 1300|| 43344|| 1503652
|-
| 36||style='text-align:center;'| 2<sup>2</sup>×3<sup>2</sup>|| 9|| 91|| 1911|| 55261|| 1813539
|-style="background-color:#ddeeff;"
| 37||style='text-align:center;'| 37|| 2|| 38|| 1370|| 50654|| 1874162
|-
| 38||style='text-align:center;'| 2×19|| 4|| 60|| 1810|| 61740|| 2215474
|-
| 39||style='text-align:center;'| 3×13|| 4|| 56|| 1700|| 61544|| 2342084
|-
| 40||style='text-align:center;'| 2<sup>3</sup>×5|| 8|| 90|| 2210|| 73710|| 2734994
|-style="background-color:#ddeeff;"
| 41||style='text-align:center;'| 41|| 2|| 42|| 1682|| 68922|| 2825762
|-
| 42||style='text-align:center;'| 2×3×7|| 8|| 96|| 2500|| 86688|| 3348388
|-style="background-color:#ddeeff;"
| 43||style='text-align:center;'| 43|| 2|| 44|| 1850|| 79508|| 3418802
|-
| 44||style='text-align:center;'| 2<sup>2</sup>×11|| 6|| 84|| 2562|| 97236|| 3997266
|-
| 45||style='text-align:center;'| 3<sup>2</sup>×5|| 6|| 78|| 2366|| 95382|| 4158518
|-
| 46||style='text-align:center;'| 2×23|| 4|| 72|| 2650|| 109512|| 4757314
|-style="background-color:#ddeeff;"
| 47||style='text-align:center;'| 47|| 2|| 48|| 2210|| 103824|| 4879682
|-
| 48||style='text-align:center;'| 2<sup>4</sup>×3|| 10|| 124|| 3410|| 131068|| 5732210
|-
| 49||style='text-align:center;'| 7<sup>2</sup>|| 3|| 57|| 2451|| 117993|| 5767203
|-
| 50||style='text-align:center;'| 2×5<sup>2</sup>|| 6|| 93|| 3255|| 141759|| 6651267
|}
{{clear}}

==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<sub>n</sub>''# 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 prime]]s, prime powers whose exponent is a power of two.<ref>{{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]</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 function|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 function|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''&nbsp;=&nbsp;''ω''(''n'') is the [[prime omega function|number of distinct prime factors]] of ''n'', ''p<sub>i</sub>'' is the ''i''th prime factor, and ''a<sub>i</sub>'' is the maximum power of ''p<sub>i</sub>'' by which ''n'' is [[divisible]], then we have: {{sfnp|Hardy|Wright|2008|pp=310 f|loc=§16.7}}

:<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''&nbsp;&ne;&nbsp;0, is equivalent to the useful formula: {{sfnp|Hardy|Wright|2008|pp=310 f|loc=§16.7}}

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

When ''x''&nbsp;=&nbsp;0, <math>\sigma_0(n)</math> is: {{sfnp|Hardy|Wright|2008|pp=310 f|loc=§16.7}}

:<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 (''p''<sub>1</sub> is 2; ''p''<sub>2</sub> is 3); noting that 24 is the product of 2<sup>3</sup>×3<sup>1</sup>, ''a''<sub>1</sub> is 3 and ''a''<sub>2</sub> 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>{{Cite arXiv |eprint = math/0411587|last1 = Euler|first1 = Leonhard|title = An observation on the sums of divisors|last2 = Bell|first2 = Jordan|year = 2004}}</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]] ({{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 <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.{{sfnp|Gioia|Vaidya|1967}}

We also note ''s''(''n'') = ''σ''(''n'')&nbsp;−&nbsp;''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 number]]s, which are the ''n'' such that ''s''(''n'') =&nbsp;''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, <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 number|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 phi|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 {{mvar|n}} 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 {{mvar|n}}, see {{OEIS2C|A005237}}.

=== Dirichlet convolutions ===
{{Main article|Dirichlet convolution}}
By definition:<math display="block">\sigma = \operatorname{Id} * \mathbf 1</math>By [[Möbius inversion formula|Möbius inversion]]:<math display="block">\operatorname{Id} = \sigma * \mu   </math>

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

:<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'')&nbsp;=&nbsp;''&sigma;''<sub>0</sub>(''n'') gives: {{sfnp|Hardy|Wright|2008|pp=326-328|loc=§17.5}}

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

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

:<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 method|Rankin–Selberg convolution]].

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

:<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}</math>

for arbitrary [[complex number|complex]] |''q''|&nbsp;≤&nbsp;1 and&nbsp;''a''.  This summation also appears as the [[Eisenstein series#Fourier series|Fourier series of the Eisenstein series]] and the [[Weierstrass elliptic functions#Invariants|invariants of the Weierstrass elliptic functions]].

For <math>k>0</math>, there is an explicit series representation with [[Ramanujan sum]]s <math> c_m(n) </math> as :<ref>{{cite book |author=E. Krätzel |title=Zahlentheorie |publisher=VEB Deutscher Verlag der Wissenschaften |place =Berlin |year=1981 |pages=130}} (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<!--linked from 'Thomas Hakon Grönwall'-->==
In [[Big O notation#Little-o notation|little-o notation]], the divisor function satisfies the inequality:{{sfnp|Apostol|1976|p=296}}{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}
:<math>\mbox{for all }\varepsilon>0,\quad d(n)=o(n^\varepsilon).</math>
More precisely, [[Severin Wigert]] showed that:{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}
:<math>\limsup_{n\to\infty}\frac{\log d(n)}{\log n/\log\log n}=\log2.</math>
On the other hand, since [[Euclid's theorem|there are infinitely many prime numbers]],{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}
:<math>\liminf_{n\to\infty} d(n)=2.</math>

In [[Big-O notation]], [[Peter Gustav Lejeune Dirichlet]] showed that the [[Average order of an arithmetic function|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}}
:<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–Mascheroni constant|Euler's gamma constant]]. Improving the bound <math>O(\sqrt{x})</math> in this formula is known as [[Divisor summatory function#Dirichlet's divisor problem|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}}
:<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 '''[[Thomas Hakon Grönwall|Grönwall]]'s theorem''', published in 1913 {{harv|Grönwall|1913}}. His proof uses [[Mertens' theorems|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'' {{harv|Ramanujan|1997}}.  The largest known value that violates the inequality is ''n''=[[5040 (number)|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 number|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:
:<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'' &gt; 1, where <math>H_n</math> is the ''n''th [[harmonic number]], {{harv|Lagarias|2002}}.

== See also ==
* [[Arithmetic function#Divisor sum convolutions|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]]

==Notes==
{{reflist}}

== References ==
*{{Citation|doi=10.4169/193009709X470128|first1=Amir|last1=Akbary|first2=Zachary|last2=Friggstad|title=Superabundant numbers and the Riemann hypothesis|url=http://webdocs.cs.ualberta.ca/~zacharyf/Papers/superabundant.pdf|journal=American Mathematical Monthly|volume=116|issue=3|year=2009|pages=273–275|url-status=dead|archive-url=https://web.archive.org/web/20140411041855/http://webdocs.cs.ualberta.ca/~zacharyf/papers/superabundant.pdf|archive-date=2014-04-11}}.
*{{Apostol IANT}}
* [[Eric Bach|Bach, Eric]]; [[Jeffrey Shallit|Shallit, Jeffrey]], ''Algorithmic Number Theory'', volume 1, 1996, MIT Press.  {{ISBN|0-262-02405-5}}, see page 234 in section 8.8.
* {{Citation | last1=Caveney | first1=Geoffrey | last2=Nicolas | first2=Jean-Louis|author2-link=Jean-Louis Nicolas | last3=Sondow | first3=Jonathan | title=Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis | url=http://www.integers-ejcnt.org/l33/l33.pdf | year=2011 | journal=INTEGERS: The Electronic Journal of Combinatorial Number Theory | volume=11 | pages=A33| bibcode=2011arXiv1110.5078C | arxiv=1110.5078 }}
*{{Citation | last1=Choie | first1=YoungJu | author1-link= YoungJu Choie | last2=Lichiardopol | first2=Nicolas | last3=Moree | first3=Pieter | author3-link=Pieter Moree | last4=Solé | first4=Patrick | title=On Robin's criterion for the Riemann hypothesis | mr=2394891 | zbl=1163.11059 | arxiv=math.NT/0604314 | year=2007 | journal=Journal de théorie des nombres de Bordeaux | issn=1246-7405 | volume=19 | issue=2 | pages=357–372 | doi=10.5802/jtnb.591| s2cid=3207238 }}
*{{citation
 | last1 = Gioia | first1 = A. A.
 | last2 = Vaidya | first2 = A. M.
 | doi = 10.2307/2315280
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2315280
 | mr = 220659
 | pages = 969–973
 | title = Amicable numbers with opposite parity
 | volume = 74
 | year = 1967| issue = 8
 }}
* {{Citation | last1=Grönwall | first1=Thomas Hakon | author1-link=Thomas Hakon Grönwall | title=Some asymptotic expressions in the theory of numbers | year=1913 | journal=Transactions of the American Mathematical Society | volume=14 | pages=113–122 | doi=10.1090/S0002-9947-1913-1500940-6| doi-access=free }}
* {{Citation | last1=Hardy | first1=G. H. | author1-link=G. H. Hardy | last2=Wright | first2=E. M. | author2-link=E. M. Wright | edition=6th | others=Revised by [[Roger Heath-Brown|D. R. Heath-Brown]] and [[Joseph H. Silverman|J. H. Silverman]]. Foreword by [[Andrew Wiles]]. | title=An Introduction to the Theory of Numbers | publisher=[[Oxford University Press]] | location=Oxford | isbn=978-0-19-921986-5 | mr=2445243 | zbl=1159.11001 | year=2008 | orig-year=1938}}
* {{citation | last=Ivić | first=Aleksandar | title=The Riemann zeta-function. The theory of the Riemann zeta-function with applications | series=A Wiley-Interscience Publication | location=New York etc. | publisher=John Wiley & Sons | year=1985 | isbn=0-471-80634-X | zbl=0556.10026 | pages=385–440 }}
* {{Citation | last1=Lagarias | first1=Jeffrey C. | author1-link=Jeffrey C. Lagarias | title=An elementary problem equivalent to the Riemann hypothesis | doi=10.2307/2695443 | jstor=2695443 | mr=1908008 | year=2002 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=109 | issue=6 | pages=534–543| arxiv=math/0008177 | s2cid=15884740 }}
* {{citation | first1 = Calvin T. | last1 = Long | year = 1972 | title = Elementary Introduction to Number Theory | edition = 2nd | publisher = [[D. C. Heath and Company]] | location = Lexington | lccn = 77171950 }}
* {{citation | first1 = Anthony J. | last1 = Pettofrezzo | first2 = Donald R. | last2 = Byrkit | year = 1970 | title = Elements of Number Theory | publisher = [[Prentice Hall]] | location = Englewood Cliffs | lccn = 77081766 }}
* {{Citation | last=Ramanujan | first=Srinivasa | author-link=Srinivasa Ramanujan | title=Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin | doi=10.1023/A:1009764017495 | mr=1606180 | year=1997 | journal=The Ramanujan Journal | issn=1382-4090 | volume=1 | issue=2 | pages=119–153| s2cid=115619659 }}
* {{Citation | last1=Robin | first1=Guy | title=Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann | mr=774171 | year=1984 | journal=[[Journal de Mathématiques Pures et Appliquées]]|series= Neuvième Série | issn=0021-7824 | volume=63 | issue=2 | pages=187–213}}
* {{citation | last=Williams | first=Kenneth S. | title=Number theory in the spirit of Liouville | zbl=1227.11002 | series=London Mathematical Society Student Texts | volume=76 | location=Cambridge | publisher=[[Cambridge University Press]] | isbn=978-0-521-17562-3 | year=2011 }}

==External links==
* {{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}}

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[[Category:Divisor function| ]]
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