Editing Arithmetic function (section)

== Multiplicative functions ==
=== ''σ''<sub>''k''</sub>(''n''), ''τ''(''n''), ''d''(''n'') – divisor sums ===
'''[[divisor function|σ<sub>''k''</sub>(''n'')]]''' is the sum of the ''k''th powers of the positive divisors of ''n'', including 1 and ''n'', where ''k'' is a complex number.

'''''σ''<sub>1</sub>(''n'')''', the sum of the (positive) divisors of ''n'', is usually denoted by '''''σ''(''n'')'''.

Since a positive number to the zero power is one, '''''σ''<sub>0</sub>(''n'')''' is therefore the number of (positive) divisors of ''n''; it is usually denoted by '''''d''(''n'')''' or '''''τ''(''n'')''' (for the German ''Teiler'' = divisors).

<math display="block">\sigma_k(n) = \prod_{i=1}^{\omega(n)} \frac{p_i^{(a_i+1)k}-1}{p_i^k-1}= \prod_{i=1}^{\omega(n)} \left(1 + p_i^k + p_i^{2k} + \cdots + p_i^{a_i k}\right).</math>

Setting ''k'' = 0 in the second product gives
<math display="block">\tau(n) = d(n) = (1 + a_{1})(1+a_{2})\cdots(1+a_{\omega(n)}).</math>

=== ''φ''(''n'') – Euler totient function ===
'''[[Euler totient function|''φ''(''n'')]]''', the Euler totient function, is the number of positive integers not greater than ''n'' that are coprime to ''n''.
<math display="block">\varphi(n) = n \prod_{p\mid n} \left(1-\frac{1}{p}\right)
= n \left(\frac{p_1 - 1}{p_1}\right)\left(\frac{p_2 - 1}{p_2}\right) \cdots \left(\frac{p_{\omega(n)} - 1}{p_{\omega(n)}}\right)
.</math>

=== ''J''<sub>''k''</sub>(''n'') – Jordan totient function ===
'''[[Jordan totient function|''J''<sub>''k''</sub>(''n'')]]''', the Jordan totient function, is the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a coprime (''k'' + 1)-tuple together with ''n''. It is a generalization of Euler's  totient, {{math|1=''φ''(''n'') = ''J''<sub>1</sub>(''n'')}}.
<math display="block">J_k(n) = n^k \prod_{p\mid n} \left(1-\frac{1}{p^k}\right)
= n^k \left(\frac{p^k_1 - 1}{p^k_1}\right)\left(\frac{p^k_2 - 1}{p^k_2}\right) \cdots \left(\frac{p^k_{\omega(n)} - 1}{p^k_{\omega(n)}}\right)
.</math>

=== ''μ''(''n'') – Möbius function===
'''[[Möbius function|''μ''(''n'')]]''', the Möbius function, is important because of the [[Möbius inversion]] formula. See ''{{slink|#Dirichlet convolution}}'', below.
<math display="block">\mu(n)=\begin{cases}
(-1)^{\omega(n)}=(-1)^{\Omega(n)} &\text{if }\; \omega(n) = \Omega(n)\\
0&\text{if }\;\omega(n) \ne \Omega(n).
\end{cases}</math>

This implies that ''μ''(1) = 1. (Because Ω(1) = ''ω''(1) = 0.)

=== ''τ''(''n'') – Ramanujan tau function ===
'''[[Ramanujan tau function|''τ''(''n'')]]''', the Ramanujan tau function, is defined by its [[generating function]] identity:
<math display="block">\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24}.</math>

Although it is hard to say exactly what "arithmetical property of ''n''" it "expresses",<ref>Hardy, ''Ramanujan'', § 10.2</ref> (''τ''(''n'') is (2''π'')<sup>−12</sup> times the ''n''th Fourier coefficient in the [[q-expansion|''q''-expansion]] of the [[Modular discriminant#Modular discriminant|modular discriminant]] function)<ref>Apostol, ''Modular Functions ...'', § 1.15, Ch. 4, and ch. 6</ref> it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain ''σ''<sub>''k''</sub>(''n'') and ''r''<sub>''k''</sub>(''n'') functions (because these are also coefficients in the expansion of [[modular form]]s).

=== ''c''<sub>''q''</sub>(''n'') – Ramanujan's sum ===
'''[[Ramanujan's sum|''c''<sub>''q''</sub>(''n'')]]''', Ramanujan's sum, is the sum of the ''n''th powers of the primitive ''q''th [[roots of unity]]:
<math display="block">c_q(n) = \sum_{\stackrel{1\le a\le q}{ \gcd(a,q)=1}} e^{2 \pi i \tfrac{a}{q} n}.</math>

Even though it is defined as a sum of complex numbers (irrational for most values of ''q''), it is an integer. For a fixed value of ''n'' it is multiplicative in ''q'':
: '''If ''q'' and ''r'' are coprime''', then <math>c_q(n)c_r(n)=c_{qr}(n).</math>

=== ''ψ''(''n'') – Dedekind psi function ===
The [[Dedekind psi function]], used in the theory of [[modular function]]s, is defined by the formula
<math display="block"> \psi(n) = n \prod_{p|n}\left(1+\frac{1}{p}\right).</math>