Some optimal links between generations of correlation averages

Abstract

For a real-valued and essentially bounded arithmetic function $f$, i.e., $f(n){\ll }_{\epsilon }{n}^{\epsilon },\forall \epsilon >0$, we \enspace give some optimal links between non-trivial bounds for the sums ${\sum }_{h\le H}{\sum }_{N<n\le 2N}f(n)f(n-h)$, ${\sum }_{N<x\le 2N}|{\sum }_{x<n\le x+H}f(n){|}^2$ and ${\sum }_{N<n\le 2N}\left|{\sum }_{0\le |n-x|\le H}\right(1-\frac{|n-x|}{H})f(n){|}^2$, with $H=o(N)$ as $N\to \infty$.

5 pages