Week 19.06.2023 – 25.06.2023

Bourgain (2015) estimated the number of prime numbers with a positive
proportion of preassigned digits in base 2. We first present a
generalization of this result to any base g at least 2. We then discuss
a more recent result for the set of squares, which may be seen as one
of the most interesting sets after primes. More precisely, for any
base g, we obtain an asymptotic formula for the number of
squares with a proportion c>0 of preassigned digits. Moreover we
provide explicit admissible values for c depending on g. Our
proof mainly follows the strategy developed by Bourgain for primes in
base 2, with new difficulties for squares. It is based on the circle
method and combines techniques from harmonic analysis together with
arithmetic properties of squares and bounds for quadratic Weyl sums.

Title: Dirichlet L-series at s = 0 and the scarcity of Euler systems

Abstract: Ever since their introduction, Euler systems have played an important role in understanding the mysterious link between analysis and arithmetic that manifests itself in leading term formulae for L-series. In this talk, I will discuss joint work with Burns, Daoud, and Seo that proves a precise description (conjectured by Coleman) of the set of all Euler systems over the rationals. This also leads to a new approach towards leading term conjectures, including a proof of the `minus part’ of the equivariant Tamagawa Number Conjecture outside 2.