Short Interval Variance and Averaged Correlations of Arithmetic Functions

Abstract

In this paper, we study the average shifted sum for general arithmetic functions by applying the standard Hardy--Littlewood circle method and using short-interval variance results. As applications, we prove some nontrivial upper bounds for shifted sums involving ${\mu }_k(n).$ Assuming the Riemann Hypothesis and the Pair Correlation Conjecture of Montgomery, we also prove similar results involving the von Mangoldt function.

Under our assumptions on the functions, the main theorem is generally less effective than the trivial bound obtained via Cauchy-Schwarz. For interested readers, the draft remains accessible, along with an explanation for the withdrawal, on the authors' website