Pillai's arithmetical function

In number theory, the gcd-sum function, also called Pillai's arithmetical function, is defined for every n by

or equivalently

where d is a divisor of n and φ is Euler's totient function.

it also can be written as

where, σ is the Divisor function, and μ is the Möbius function.

This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.