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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{| class="wikitable" style="text-align:right; float:left"

! ''n'' !! prime factorization !! {{sigma}}₀(''n'')!! {{sigma}}₁(''n'')!! {{sigma}}₂(''n'')!! {{sigma}}₃(''n'')!! {{sigma}}₄(''n'')

|-

| 1||style='text-align:center;'| 1|| 1|| 1|| 1|| 1|| 1

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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>

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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==

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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} = \sum_{n=1}^\infty \operatorname{Li}_{-a}(q^n)</math>

for arbitrary [[complex number|complex]] |''q''|&nbsp;≤&nbsp;1 and&nbsp;''a'' (<math>\operatorname{Li}</math> is the [[polylogarithm]]). 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>

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| 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 | issue=1 | 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 }}