Editing Arithmetic function (section)

=== ''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>