In number theory, the totient summatory function is a summatory function of Euler's totient function defined by:
It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.
Properties
edit
Using Möbius inversion to the totient function, we obtain
-
Φ(n) has the asymptotic expansion
-
where ζ(2) is the Riemann zeta function for the value 2.
Φ(n) is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.
The summatory of reciprocal totient function
edit
The summatory of reciprocal totient function is defined as
-
Edmund Landau showed in 1900 that this function has the asymptotic behavior
-
where γ is the Euler–Mascheroni constant,
-
and
-
The constant A = 1.943596... is sometimes known as Landau's totient constant. The sum is convergent and equal to:
-
In this case, the product over the primes in the right side is a constant known as totient summatory constant,[1] and its value is:
-
See also
edit
References
edit
External links
edit