Andrew Granville's Home Page

Preprints accepted for publication

Sieve weights and their smoothings
(with Dimitris Koukoulopoulos and James Maynard)
Annales Scientifiques de l'École Normale Supérieure

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''.


An alternative to Vaughan's identity
Rivista di Matematica della Università di Parma

We exhibit an identity that plays the same role as Vaughan's identity but is arguably simpler.