Three conjectures about character sums

Abstract

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L(1,chi ) and L(1+it,chi )) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.