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Preprints submitted for publication

Sieve weights and their smoothings
(with Dimitris Koukoulopoulos and James Maynard)

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''.

Part-and-parcel of the study of ``multiplicative number theory'' is the study of the distribution of multiplicative functions \( f\) in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we prove that such a result has been so elusive because \( f \) can be ``pretentious'' in two different ways. Firstly it might correlate with a character of small conductor, which can be ruled out by assuming a ``Siegel-Walfisz'' type criterion for \( f\). Secondly \( f\) might be particularly awkward on large primes, and this can be avoided by restricting our attention to smoothly supported \( f\). Under these assumptions we recover a Bombieri-Vingradov Theorem for multiplicative \( f\). For a fixed residue class \( a\) we extend such averages out to moduli \( \leq x^{\frac {20}{39}-\delta}\) .