Sieve weights and their smoothings

TL;DR

This paper analyzes the asymptotic behavior of moments of smoothed divisor sums of the Möbius function, revealing how the main contributions shift depending on the level of smoothing and the size of moments, with implications for sieve weights.

Contribution

It provides a detailed asymptotic analysis of divisor sum moments, identifying thresholds where the main contributions change, and explores analogous phenomena in finite fields and permutations.

Findings

Main contributions shift from large prime factors to many prime factors as moments increase.

Threshold for 'small' moments depends on the binomial coefficient and smoothing level.

Different behaviors observed in finite fields and permutations, explained via Dirichlet characters.

Abstract

We obtain asymptotic formulas for the $2k$th moments of partially smoothed divisor sums of the M\"obius function. When $2k$ is small compared with $A$, the level of smoothing, then the main contribution to the moments come from integers with only large prime factors, as one would hope for in sieve weights. However if $2k$ is any larger, compared with $A$, then the main contribution to the moments come from integers with quite a few prime factors, which is not the intention when designing sieve weights. The threshold for "small" occurs when $A=\frac 1{2k} \binom{2k}{k}-1$. One can ask analogous questions for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, with even less cancellation in the divisor sums. We give, we hope, a plausible explanation for this phenomenon, by studying the analogous sums for Dirichlet characters, and obtaining each type of behaviour depending on whether or not the character is "exceptional".