Editing Root of unity (section)

==Summation==
Let {{math|SR(''n'')}} be the sum of all the {{mvar|n}}th roots of unity, primitive or not. Then

:<math>\operatorname{SR}(n) =
\begin{cases}
1, & n=1\\
0, & n>1.
\end{cases}</math>

This is an immediate consequence of [[Vieta's formulas]]. In fact, the {{mvar|n}}th roots of unity being the roots of the polynomial {{math|''X''<sup>&thinsp;''n''</sup> − 1}}, their sum is the [[coefficient]] of degree {{math|''n'' − 1}}, which is either 1 or 0 according whether {{math|1=''n'' = 1}} or {{math|''n'' > 1}}.

Alternatively, for {{math|1=''n'' = 1}} there is nothing to prove, and for {{math|1=''n'' > 1}} there exists a root {{math|''z'' ≠ 1}} – since the set {{math|''S''}} of all the {{mvar|n}}th roots of unity is a [[group (mathematics)|group]], {{math|1=''z{{space|hair}}S'' =  ''S''}}, so the sum satisfies {{math|1= ''z'' SR(''n'') =  SR(''n'')}}, whence {{math|1=SR(''n'') = 0}}.

Let {{math|SP(''n'')}} be the sum of all the primitive {{mvar|n}}th roots of unity. Then

:<math>\operatorname{SP}(n) = \mu(n),</math>

where {{math|''μ''(''n'')}} is the [[Möbius function]].

In the section [[Root of unity#Elementary properties|Elementary properties]], it was shown that if {{math|R(''n'')}} is the set of all {{mvar|n}}th roots of unity and {{math|P(''n'')}} is the set of primitive ones, {{math|R(''n'')}} is a disjoint union of the {{math|P(''n'')}}:

:<math>\operatorname{R}(n) = \bigcup_{d \,|\, n}\operatorname{P}(d),</math>

This implies

:<math>\operatorname{SR}(n) = \sum_{d \,|\, n}\operatorname{SP}(d).</math>

Applying the [[Möbius inversion formula]] gives

:<math>\operatorname{SP}(n) = \sum_{d \,|\, n}\mu(d)\operatorname{SR}\left(\frac{n}{d}\right).</math>

In this formula, if {{math|''d'' < ''n''}}, then {{math|1=SR({{sfrac|''n''|''d''}}) = 0}}, and for {{math|1=''d'' = ''n''}}: {{math|1=SR({{sfrac|''n''|''d''}}) = 1}}.  Therefore, {{math|1=SP(''n'') = ''μ''(''n'')}}.

This is the special case {{math|''c''<sub>''n''</sub>(1)}} of [[Ramanujan's sum]] {{math|''c''<sub>''n''</sub>(''s'')}},<ref name="apostol">{{cite book
|last = Apostol |first = Tom M. |author-link = Tom M. Apostol
|year = 1976
|title = Introduction to Analytic Number Theory
|series = Undergraduate Texts in Mathematics |url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA160
|page = 160
|publisher = Springer
|doi = 10.1007/978-1-4757-5579-4|isbn = 978-1-4419-2805-4 }}</ref> defined as the sum of the {{mvar|s}}th powers of the primitive {{mvar|n}}th roots of unity:

:<math>c_n(s) = \sum_{a = 1 \atop \gcd(a, n) = 1}^n e^{2 \pi i \frac{a}{n} s}.</math>