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!
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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).
