Andrew Granville's Home Page

Preprints still in development

We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is ``pretentious''. This leads to applications in the circle method, and a natural interpretation of the local-global principle.


Halasz in short intervals
(with Adam Harper, Kaisa Matomaki, and Maksym Radziwill)

We show Halasz's Theorem in intervals of length \( x^{7/12} \) (removing integers with a prime factor larger than the length of the interval). The method extends to intervals of length \( >x^{1/2+\epsilon} \) but restricting to \( y\)-smooth integers where \( y\) is significantly smaller than the length of the interval.

In this article we study the similarities in the anatomies of integers and permutations.