Divisor function: Difference between revisions

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{{short description|Arithmetic function related to the divisors of an integer}}

[[Image:Divisor.svg|thumb|right|Divisor function ''σ''₀(''n'') up to ''n'' = 250]]

[[Image:Sigma function.svg|thumb|right|Sigma function ''σ''₁(''n'') up to ''n'' = 250]]

[[Image:Divisor square.svg|thumb|right|Sum of the squares of divisors, ''σ''₂(''n''), up to ''n'' = 250]]

[[Image:Divisor cube.svg|thumb|right|Sum of cubes of divisors, ''σ''₃(''n'') up to ''n'' = 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]].

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==Definition==

The '''sum of positive divisors function''' ''σ''_''z''(''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''₀(''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''₁(''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''₁(''n'')&nbsp;&minus;&nbsp;''n''; the [[aliquot sequence]] of ''n'' is formed by repeatedly applying the aliquot sum function.

==Example==

For example, ''σ''₀(12) is the number of the divisors of 12:

: <math>

\begin{align}

\sigma_{0}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''₁(12) is the sum of all the divisors:

: <math>

\begin{align}

\sigma_{1}sigma_1(12) & = 1^1 + 2^1 + 3^1 + 4^1 + 6^1 + 12^1 \\

& = 1 + 2 + 3 + 4 + 6 + 12 = 28,

\end{align}

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</math>

''σ''_-1−1(''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}

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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''₁'' is 2; ''p''₂'' is 3); noting that 24 is the product of 2³×3¹, ''a''₁ is 3 and ''a''₂ 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>