Vaughan identity for the Möbius function ($\mu ={\mu }_{\le U}+{\mu }_{\le V}-{\mu }_{\le U}\ast {\mu }_{\le V}\ast 1+{\mu }_{>U}\ast {\mu }_{>V}\ast 1$)

The displayed identity is exact for all positive integers and all positive cutoffs. Expand $(\mu -{\mu }_{\le U})\ast (\mu -{\mu }_{\le V})\ast 1$ and use $\mu \ast 1=\epsilon$. The two truncated terms are short sums; the negative term is a type I sum over divisors at most $UV$; the last term is a type II sum with both factor variables larger than a cutoff. This form of the Vaughan identity is specific to Möbius weights.

For $|f|\le 1$, if $|{\sum }_{n\le X}\mu (n)f(n)|\ge \delta X$ and $U+V\le \delta X/2$, the Vaughan identity for the Möbius function gives a type I sum or a type II sum of modulus at least $\delta X/4$. The first is ${\sum }_{d\le UV}{c}_d{\sum }_{k\le X/d}f(dk)$, with $c_d={\sum }_{bc=d,b\le U,c\le V}\mu (b)\mu (c)$. The second is ${\sum }_{d>V,w>U,dw\le X}{a}_d\mu (w)f(dw)$, with $a_d={\sum }_{c\mid d,c>V}\mu (c)$. Both coefficient sequences are bounded by the divisor function. Thus correlation forces either detection by small-modulus periodic indicators or a large multiplicatively organized bilinear sum, without claiming that $f$ itself is literally periodic or multiplicative.