totient summatory function

Input interpretation

totient summatory function

Alternate names

totient constant | totient function constants

Definition

The summatory function Φ(n) of the totient function ϕ(n) is defined by Φ(n) | congruent | sum_(k = 1)^n ϕ(k) | = | sum_(m = 1)^n m sum_(d|m) (μ(d))/d | = | sum_(d = 1)^n μ(d) sum_(d' = 1)^(⌊n/d⌋) d' | = | 1/2 sum_(d = 1)^n μ(d)⌊n/d ⌋(1 + ⌊n/d ⌋) (Hardy and Wright 1979, p. 268), plotted as the red curve above. The first values of Φ(n) are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (OEIS A002088).
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