Vaughn's identity is a useful way to decompose the von Mangoldt function $\Lambda(n)$ into Type I and Type II components, and this is used in many problems involving prime numbers. I was wondering if a similar decomposition was available for other arithmetic functions as well or not. For example, I was wondering is there one available for the function $f(n) = n^{it}$ or the Mobius function? Any comments or references are appreciated. Thank you very much!



Are there Vaughn's identity type decompositions for other arithmetic functions?
Proposition 13.5 of the book Analytic Number Theory by Iwaniec and Kowalski gives a decomposition for the Mobius function: let $y, z\ge1$. Then for any $m>\max\{y, z\}$, we have
$$
\mu(m)=-\mathop{\sum\sum}\_{\substack{bc|m\\b\le y,c\le z}}\mu(b)\mu(c)+\mathop{\sum\sum}\_{\substack{bc|m\\b>y,c>z}}\mu(b)\mu(c).
$$
This Proposition can be used to prove the following analogue of Bombieri-Vinogradov Theorem:
$$
\sum\_{q\le Q}\max\_a\bigg|\sum\_{\substack{m\le x\\m\equiv a\pmod q}}\mu(m)\bigg|\ll x(\log x)^{-A}
$$
for any $A>0$, where $Q=x^{1/2}(\log x)^{-B}$ with $B=B(A)$ and the implied constant depends only on $A$ (See (17.10) of the same book).


