totient constant

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).

prime products | totient function | totient valence function