Pillai's arithmetical function

In number theory, the gcd-sum function,^[1] also called Pillai's arithmetical function,^[1] is defined for every ${\displaystyle n}$ by

${\displaystyle P(n)=\sum _{k=1}^{n}\gcd(k,n)}$

or equivalently^[1]

${\displaystyle P(n)=\sum _{d\mid n}d\varphi (n/d)}$

where ${\displaystyle d}$ is a divisor of ${\displaystyle n}$ and ${\displaystyle \varphi }$ is Euler's totient function.

it also can be written as^[2]

${\displaystyle P(n)=\sum _{d\mid n}d\tau (d)\mu (n/d)}$

where, ${\displaystyle \tau }$ is the divisor function, and ${\displaystyle \mu }$ is the Möbius function.

This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.^[3]

^[4]