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, χ) and L(1 + it, χ)) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

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