= Type I–type II inverse principle for Möbius correlation
For $|f|\leq1$, if $|\sum_{n\leq X}\mu(n)f(n)|\geq\delta X$ and $U+V\leq\delta X/2$, 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 $\sum_{d\leq UV}c_d\sum_{k\leq X/d}f(dk)$, with $c_d=\sum_{bc=d,b\leq U,c\leq V}\mu(b)\mu(c)$. The second is $\sum_{d>V,w>U,dw\leq X}a_d\mu(w)f(dw)$, 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.
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