Totient summatory function
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by:Properties
Using Möbius inversion to the totient function, we obtain
has the asymptotic expansion
where is the Riemann zeta function for the value 2.The summatory of reciprocal totient function
The summatory of reciprocal totient function is defined as
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 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, and its value is:
