Editing Möbius function (section)

== Mertens function ==
In number theory another [[arithmetic function]] closely related to the Möbius function is the [[Mertens function]], defined by

:<math>M(n) = \sum_{k = 1}^n \mu(k)</math>

for every natural number {{mvar|n}}. This function is closely linked with the positions of zeroes of the [[Riemann zeta function]]. See the article on the [[Mertens conjecture]] for more information about the connection between <math>M(n)</math> and the [[Riemann hypothesis]].

From the formula

:<math>\mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,n)=1}} e^{2\pi i \frac{k}{n}},</math>

it follows that the Mertens function is given by

:<math>M(n)= -1+\sum_{a\in \mathcal{F}_n} e^{2\pi i a},</math>

where <math>\mathcal{F}_n</math> is the [[Farey sequence]] of order <math>n</math>.

This formula is used in the proof of the [[Farey sequence#Riemann hypothesis|Franel–Landau theorem]].{{sfn|Edwards|1974|loc=Ch. 12.2}}