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| [[Image:Divisor.svg|thumb|right|Divisor function σ<sub>0</sub>(''n'') up to ''n'' = 250]]
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| [[Image:Sigma function.svg|thumb|right|Sigma function σ<sub>1</sub>(''n'') up to ''n'' = 250]]
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| [[Image:Divisor square.svg|thumb|right|Sum of the squares of divisors, σ<sub>2</sub>(''n''), up to ''n'' = 250]]
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| [[Image:Divisor cube.svg|thumb|right|Sum of cubes of divisors, σ<sub>3</sub>(''n'') up to ''n'' = 250]]
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| 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''. 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]].
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| A related function is the [[divisor summatory function]], which, as the name implies, is a sum over the divisor function.
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| ==Definition==
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| The '''sum of positive divisors function''' σ<sub>''x''</sub>(''n''), for a real or complex number ''x'', is defined as the [[sum]] of the ''x''th [[Exponentiation|powers]] of the positive [[divisor]]s of ''n''. It can be expressed in [[Summation#Capital-sigma notation|sigma notation]] as
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| :<math>\sigma_{x}(n)=\sum_{d|n} d^x\,\! ,</math>
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| where <math>{d|n}</math> is shorthand for "''d'' [[divides]] ''n''".
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| 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> {{OEIS|id=A000005}}. When ''x'' 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 equivalent to σ<sub>1</sub>(''n'') ({{OEIS2C|id=A000203}}).
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| 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'') − ''n''; the [[aliquot sequence]] of ''n'' is formed by repeatedly applying the aliquot sum function. | |
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| ==Example==
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| For example, σ<sub>0</sub>(12) is the number of the divisors of 12:
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| : <math>
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| \begin{align}
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| \sigma_{0}(12) & = 1^0 + 2^0 + 3^0 + 4^0 + 6^0 + 12^0 \\
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| & = 1 + 1 + 1 + 1 + 1 + 1 = 6,
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| \end{align}
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| </math>
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| while σ<sub>1</sub>(12) is the sum of all the divisors:
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| : <math>
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| \begin{align}
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| \sigma_{1}(12) & = 1^1 + 2^1 + 3^1 + 4^1 + 6^1 + 12^1 \\
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| & = 1 + 2 + 3 + 4 + 6 + 12 = 28,
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| \end{align}
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| </math>
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| and the aliquot sum s(12) of proper divisors is:
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| : <math>
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| \begin{align}
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| s(12) & = 1^1 + 2^1 + 3^1 + 4^1 + 6^1 \\
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| & = 1 + 2 + 3 + 4 + 6 = 16.
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| \end{align}
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| </math>
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| ==Table of values==
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| {| class="wikitable"
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| |-
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| ! ''n''
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| ! Divisors
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| ! σ<sub>0</sub>(''n'')
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| ! σ<sub>1</sub>(''n'')
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| ! ''s''(''n'') = σ<sub>1</sub>(''n'') − ''n''
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| ! Comment
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| |-
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| ! 1
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| | 1
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| | 1
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| | 1
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| | 0
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| | square number: σ<sub>0</sub>(''n'') is odd; power of 2: s(''n'') = ''n'' − 1 (almost-perfect)
| |
| |-
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| ! 2
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| | 1,2
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| | 2
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| | 3
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| | 1
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| | Prime: σ<sub>1</sub>(n) = 1+n so s(n) =1
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| |-
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| ! 3
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| | 1,3
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| | 2
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| | 4
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| | 1
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| | Prime: σ<sub>1</sub>(n) = 1+n so s(n) =1
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| |-
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| ! 4
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| | 1,2,4
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| | 3
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| | 7
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| | 3
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| | square number: σ<sub>0</sub>(''n'') is odd; power of 2: ''s''(''n'') = ''n'' − 1 (almost-perfect)
| |
| |-
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| ! 5
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| | 1,5
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| | 2
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| | 6
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| | 1
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| | Prime: σ<sub>1</sub>(n) = 1+n so s(n) =1
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| |-
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| ! 6
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| | 1,2,3,6
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| | 4
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| | 12
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| | 6
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| | first [[perfect number]]: ''s''(''n'') = ''n''
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| |-
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| ! 7
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| | 1,7
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| | 2
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| | 8
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| | 1
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| | Prime: σ<sub>1</sub>(n) = 1+n so s(n) =1
| |
| |-
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| ! 8
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| | 1,2,4,8
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| | 4
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| | 15
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| | 7
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| | power of 2: ''s''(''n'') = ''n'' − 1 (almost-perfect)
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| |-
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| ! 9
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| | 1,3,9
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| | 3
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| | 13
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| | 4
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| | square number: σ<sub>0</sub>(''n'') is odd
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| |-
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| ! 10
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| | 1,2,5,10
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| | 4
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| | 18
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| | 8
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| |
| |
| |-
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| ! 11
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| | 1,11
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| | 2
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| | 12
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| | 1
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| | Prime: σ<sub>1</sub>(n) = 1+n so s(n) =1
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| |-
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| ! 12
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| | 1,2,3,4,6,12
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| | 6
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| | 28
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| | 16
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| | first [[abundant number]]: ''s''(''n'') > ''n''
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| |-
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| ! 13
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| | 1,13
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| | 2
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| | 14
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| | 1
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| | Prime: σ<sub>1</sub>(n) = 1+n so s(n) =1
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| |-
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| ! 14
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| | 1,2,7,14
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| | 4
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| | 24
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| | 10
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| |
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| |-
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| ! 15
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| | 1,3,5,15
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| | 4
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| | 24
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| | 9
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| |-
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| ! 16
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| | 1,2,4,8,16
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| | 5
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| | 31
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| | 15
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| | square number: σ<sub>0</sub>(''n'') is odd; power of 2: ''s''(''n'') = ''n'' − 1 (almost-perfect)
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| |}
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| The cases {{math|x{{=}}2}}, {{math|x{{=}}3}} and so on are tabulated
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| in {{OEIS2C|A001157}}, {{OEIS2C|A001158}}, {{OEIS2C|A001159}},
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| {{OEIS2C|A001160}}, {{OEIS2C|A013954}}, {{OEIS2C|A013955}} ...
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| ==Properties==
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| For a non-square integer every divisor d of n is paired with divisor n/d of n and <math>\sigma_{0}(n)</math> is then 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 then odd.
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| For a [[prime number]] ''p'',
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| :<math>
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| \begin{align}
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| \sigma_0(p) & = 2 \\
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| \sigma_0(p^n) & = n+1 \\
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| \sigma_1(p) & = p+1
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| \end{align}
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| </math>
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| because by definition, the factors of a prime number are 1 and itself. Also,where ''p<sub>n</sub>''# denotes the [[primorial]],
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| :<math> \sigma_0(p_n\#) = 2^n \, </math>
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| since ''n'' prime factors allow a sequence of binary selection (<math>p_{i}</math> or 1) from ''n'' terms for each proper divisor formed.
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| Clearly, <math>1 < \sigma_0(n) < n</math> and σ(''n'') > ''n'' for all ''n'' > 2.
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| The divisor function is [[multiplicative function|multiplicative]], but not [[Completely multiplicative function|completely multiplicative]]. The consequence of this is that, if we write
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| :<math>n = \prod_{i=1}^r p_i^{a_i}</math>
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| where ''r'' = ''ω''(''n'') is the number of distinct [[prime factor]]s 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
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| :<math>\sigma_x(n) = \prod_{i=1}^{r} \frac{p_{i}^{(a_{i}+1)x}-1}{p_{i}^x-1}</math>
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| which is equivalent to the useful formula:
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| :<math>
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| \sigma_x(n) = \prod_{i=1}^r \sum_{j=0}^{a_i} p_i^{j x} =
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| \prod_{i=1}^r (1 + p_i^x + p_i^{2x} + \cdots + p_i^{a_i x}).
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| </math>
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| It follows (by setting ''x'' = 0) that ''d''(''n'') is:
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| :<math>\sigma_0(n)=\prod_{i=1}^r (a_i+1).</math>
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| 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:
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| : <math>
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| \begin{align}
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| \sigma_0(24) & = \prod_{i=1}^{2} (a_i+1) \\
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| & = (3 + 1)(1 + 1) = 4 \times 2 = 8.
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| \end{align}
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| </math>
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| The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.
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| We also note ''s''(''n'') = ''σ''(''n'') − ''n''. Here ''s''(''n'') denotes the sum of the proper divisors of ''n'', i.e. the divisors of ''n'' excluding ''n'' itself.
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| This function is the one used to recognize [[perfect number]]s 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]].
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| If n is a power of 2, e.g. <math>n = 2^k</math>, then <math>\sigma(n) = 2 \times 2^k - 1 = 2n - 1,</math> and ''s(n) = n - 1'', which makes ''n'' [[Almost perfect number|almost-perfect]].
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| As an example, for two distinct primes ''p'' and ''q'' with ''p < q'', let
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| :<math>n = pq. \, </math>
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| Then
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| :<math>\sigma(n) = (p+1)(q+1) = n + 1 + (p+q), \, </math>
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| :<math>\varphi(n) = (p-1)(q-1) = n + 1 - (p+q), \, </math>
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| and
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| :<math>n + 1 = (\sigma(n) + \varphi(n))/2, \, </math>
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| :<math>p + q = (\sigma(n) - \varphi(n))/2, \, </math>
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| where ''φ''(''n'') is [[Euler phi|Euler's totient function]].
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| Then, the roots of:
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| :<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>
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| allows us to express ''p'' and ''q'' in terms of ''σ''(''n'') and ''φ''(''n'') only, without even knowing ''n'' or ''p+q'', as:
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| :<math>p = (\sigma(n) - \varphi(n))/4 - \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}, \, </math>
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| :<math>q = (\sigma(n) - \varphi(n))/4 + \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}. \, </math>
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| 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''.
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| In 1984, [[Roger Heath-Brown]] proved that
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| :<math>\sigma_0(n) = \sigma_0(n + 1)</math>
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| will occur infinitely often.
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| ==Series relations==
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| Two [[Dirichlet series]] involving the divisor function are:
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| :<math>\sum_{n=1}^\infty \frac{\sigma_{a}(n)}{n^s} = \zeta(s) \zeta(s-a),</math>
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| which for ''d''(''n'') = ''σ''<sub>0</sub>(''n'') gives
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| : <math>\sum_{n=1}^\infty \frac{d(n)}{n^s} = \zeta^2(s),</math>
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| and
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| :<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>
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| A [[Lambert series]] involving the divisor function is:
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| :<math>\sum_{n=1}^\infty q^n \sigma_a(n) = \sum_{n=1}^\infty \frac{n^a q^n}{1-q^n}</math>
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| for arbitrary [[complex number|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]].
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| ==Approximate growth rate==
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| In [[Big O notation#Little-o notation|little-o notation]], the divisor function satisfies the inequality (see page 296 of Apostol’s book<ref name="Apostol">{{Apostol IANT}}</ref>)
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| :<math>\mbox{for all }\epsilon>0,\quad d(n)=o(n^\epsilon).</math>
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| More precisely, [[Severin Wigert]] showed that
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| :<math>\limsup_{n\to\infty}\frac{\log d(n)}{\log n/\log\log n}=\log2.</math>
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| On the other hand, since [[Prime number#There are infinitely many prime numbers|there are infinitely many prime numbers]],
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| :<math>\liminf_{n\to\infty} d(n)=2.</math>
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| 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 (see Theorem 3.3 of Apostol’s book<ref name="Apostol"/>)
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| :<math>\mbox{for all } x\geq1, \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}),</math>
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| where <math>\gamma</math> is [[Euler–Mascheroni constant|Euler's 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]]
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| {{anchor|Robin's theorem|Robin's inequality|Grönwall's theorem}}
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| The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by:
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| :<math>
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| \limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma,
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| </math>
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| 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' 3rd theorem]], which says that
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| :<math>\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^{\gamma},</math>
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| where ''p'' denotes a prime.
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| In 1915, Ramanujan proved that under the assumption of the [[Riemann hypothesis]], the inequality:
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| :<math>\ \sigma(n) < e^\gamma n \log \log n </math> (Robin's inequality)
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| holds for all sufficiently large ''n'' {{harv|Ramanujan|1997}}. In 1984 [[Guy Robin]] proved that the inequality is true for all ''n'' ≥ 5,041 if and only if the Riemann hypothesis is true {{harv|Robin|1984}}. This is '''Robin's theorem''' and the inequality became known after him. The largest known value that violates the inequality is ''n''=5,040. If the Riemann hypothesis is true, there are no greater exceptions. If the hypothesis is false, then Robin showed there are an infinite number of values of ''n'' that violate the inequality, and it is known that the smallest such ''n'' ≥ 5,041 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}}.
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| A related bound was given by [[Jeffrey Lagarias]] in 2002, who proved that the Riemann hypothesis is equivalent to the statement that
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| :<math> \sigma(n) \le H_n + \ln(H_n)e^{H_n}</math> | |
| for every [[natural number]] ''n'', where <math>H_n</math> is the ''n''th [[harmonic number]], {{harv|Lagarias|2002}}.
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| Robin also proved, unconditionally, that the inequality
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| :<math>\ \sigma(n) < e^\gamma n \log \log n + \frac{0.6483\ n}{\log \log n}</math>
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| holds for all ''n'' ≥ 3.
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| == See also ==
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| * [[Euler's totient function]] (Euler's phi function)
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| * [[Table of divisors]]
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| * [[Arithmetic function#Divisor sum convolutions|Divisor sum convolutions]] Lists a few identities involving the divisor functions
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| * [[Unitary divisor]]
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| ==Notes==
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| <references/>
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| == 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}}.
| |
| * [[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 | 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}}
| |
| *{{Citation | last1=Choie | first1=YoungJu | last2=Lichiardopol | first2=Nicolas | last3=Moree | first3=Pieter | last4=Solé | first4=Patrick | title=On Robin's criterion for the Riemann hypothesis | url=http://jtnb.cedram.org/item?id=JTNB_2007__19_2_357_0 | mr=2394891 |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}}
| |
| * {{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}}
| |
| * {{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}}
| |
| * {{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 = 77-171950 }}
| |
| * {{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}}
| |
| * {{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 = 77-81766 }}
| |
| * {{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}}
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| * {{mathworld|urlname=DivisorFunction|title=Divisor Function}}
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| * {{mathworld|urlname=RobinsTheorem|title=Robin's Theorem}}
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| * [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.
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| {{Divisor classes}}
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| [[Category:Divisor function| ]]
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| [[Category:Number theory]]
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| {{Link FA|hu}}
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| [[hu:Osztóösszeg-függvény]]
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| [[pl:Funkcja σ]]
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