Editing Euler's totient function (section)

==Generating functions==

The [[Dirichlet series]] for {{math|''φ''(''n'')}} may be written in terms of the [[Riemann zeta function]] as:<ref>{{harvnb|Hardy|Wright|1979|loc=thm. 288}}</ref>
:<math>\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}</math>
where the left-hand side converges for <math>\Re (s)>2</math>.

The [[Lambert series]] generating function is<ref>{{harvnb|Hardy|Wright|1979|loc=thm. 309}}</ref>

:<math>\sum_{n=1}^{\infty} \frac{\varphi(n) q^n}{1-q^n}= \frac{q}{(1-q)^2}</math>

which converges for {{math|{{abs|''q''}} < 1}}.

Both of these are proved by elementary series manipulations and the formulae for {{math|''φ''(''n'')}}.