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

or equivalently

where  is a divisor of  and  is Euler's totient function. it also can be written as

where,  is the divisor function, and  is the Möbius function. This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933. 