Totient summatory function

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, where.
The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32,.... Values for powers of 10 are 1, 32, 3044, 304192, 30397486, 3039650754,....

Properties

Applying Möbius inversion to the totient function yields
where is the Möbius function. has the asymptotic expansion
where is the Riemann zeta function evaluated at 2, which is.

Reciprocal totient summatory function

The summatory function of the reciprocal of the totient is
Edmund Landau showed in 1900 that this function has the asymptotic behavior
where is the Euler–Mascheroni constant,
and
The constant is sometimes known as Landau's totient constant. The sum converges to
In this case, the product over the primes in the right side is a constant known as the totient summatory constant, and its value is