Andrew Granville's Home Page

Preprints accepted for publication

Beyond the LSD method for the partial sums of multiplicative functions
(with Dimitris Koukoulopoulos)
to appear in the Ramanujan Journal

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.

Article

When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?
(with Xuancheng (Fernando) Shao)
to appear in the Forum of Mathematics, Sigma

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.

Article

A new proof of Halasz's Theorem, and its consequences
(with Adam Harper and K. Soundararajan)
to appear in Compositio Mathematica

Halasz'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).

Article

Natural exact covering systems and the reversion of the M\"obius series
(with Ian Goulden, L. Bruce Richmond and Jeffrey Shallit)
to appear in the Ramanujan Journal

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 \( a_n\) .

Article