(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''.

(with Xuancheng (Fernando) Shao)

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 \( \leq x^{\frac {20}{39}-\delta}\) .

(with Xuancheng (Fernando) Shao)

Let \( f\) and \( g\) be \( 1\)-bounded multiplicative functions for which \( f*g=1_{.=1}\). The Bombieri-Vinogradov Theorem holds for both \( f\) and \( g\)
if and only if
the Siegel-Walfisz criterion holds for both \( f\) and \( g\), and
the Bombieri-Vinogradov Theorem holds for \( f\) restricted to the primes.

(with Adam Harper and K. Soundararajan)

Halász's Theorem gives an upper bound for the mean value of a multiplicative function \( f\). The bound is sharp for general such \( f\), and, in particular, it implies that a multiplicative function with \( |f(n)|\le 1\) has either mean value \( 0\), or is ``close to" \( n^{it}\) for some fixed \( t\). The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to short intervals and to arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel's Theorem), and that there are always primes near to the start of an arithmetic progression (Linnik's Theorem).

(with Adam Harper and K. Soundararajan)

We prove a sharp version of Halász's theorem on sums \( \sum_{n \leq x} f(n)\) of multiplicative functions \( f\) with \( |f(n)|\le 1\). Our proof avoids the ``average of averages'' and ``integration over \( \alpha\)'' manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method we express \( \sum_{n \leq x} f(n)\) as a triple Dirichlet convolution, and apply Perron's formula.

(with Dimitris Koukoulopoulos)

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function \( f\) whose prime values are \(\alpha\) on average. In the literature, the average is usually taken to be \(\alpha\) with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of \(f\) , developing new techniques to do so.

(with Ian Goulden, L. Bruce Richmond and Jeffrey Shallit)

Let \( a_n\) denote the number of natural exact covering systems of cardinality \( n \) and let \( A(x)=\sum_{n\geq 1} a_nx^n.\) If \( M(x)=\sum_{n\geq 1} \mu(n)x^n\)
then \( A(M(x))=M(A(x))=x.\) Using this result, we deduce an asymptotic expression for \( a_n\) .