In number theory, the totient summatory function
is a summatory function of Euler's totient function defined by It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n. The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, ... (sequence A002088 in the OEIS).
Applying Möbius inversion to the totient function yields Φ(n) has the asymptotic expansion where ζ(2) is the Riemann zeta function evaluated at 2, which is
[1] The summatory function of the reciprocal of the totient is Edmund Landau showed in 1900 that this function has the asymptotic behavior[citation needed] where γ is the Euler–Mascheroni constant, and The constant A = 1.943596... is sometimes known as Landau's totient constant.
converges to In this case, the product over the primes in the right side is a constant known as the totient summatory constant,[2] and its value is