Definition:Ramanujan Sum
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Definition
Let $e: \R \to \C$ be the mapping defined as:
- $\forall \alpha \in \R: \map e \alpha := \map \exp {2 \pi i \alpha}$
For $q \in \N_{>0}$, $n \in \N$, the Ramanujan sum is defined as:
- $\ds \map {c_q} n = \sum_{\substack {1 \mathop \le a \mathop \le q \\ \gcd \set {a, q} \mathop = 1} } \map e {\frac {a n} q}$
Also see
By Condition for Complex Root of Unity to be Primitive, $\map {c_q} n$ is the sum of the $n$th powers of the primitive $q$th roots of unity.
This result is not to be confused with Ramanujan Summation.
Source of Name
This entry was named for Srinivasa Aiyangar Ramanujan.