2025-26

Abstract: I will discuss some new results on moments of zeta'(rho), the derivative of the Riemann zeta function evaluated at the zeta zeros. Despite being a complex function evaluated at complex points, it turns out to be real and positive on average. We will discuss this from both theoretical and heuristic viewpoints.

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Abstract: The periodic points of a discrete algebraic dynamical system control its local and global dynamical behaviour. When we impose an arithmetic structure on such a system, we do not generally expect periodic points to be rational. The central open conjecture in arithmetic dynamics asks whether this arithmetic structure imposes uniform constraints on the possible periods of points for families of algebraic dynamical systems. In this talk, we will discuss this conjecture, how it generalizes the torsion conjecture—in particular, the celebrated theorems of Mazur and Merel on rational torsion of elliptic curves—and survey some recent progress on and strategies for attacking this problem.

Abstract: A classical result of Selberg says that \log|\zeta(1/2 + it)| has a Gaussian limit distribution. We expect the same thing holds for \log|L(1/2, \chi)| for \chi being over the primitive Dirichlet characters modulo q, as q tends to infinity. Proving such a result remains completely out of reach, as it would imply 100% of these central L-values are non-zero, which is a well-known open conjecture. In this talk, I will describe how one can establish a weighted central limit theorem for the central values of Dirichlet L-functions. Under the Generalized Riemann Hypothesis, one can also obtain a weighted central limit theorem for the joint distribution of the central L-values corresponding to twists of two distinct primitive Hecke eigenforms. This is joint work with Natalie Evans, Stephen Lester and Kyle Pratt.

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Abstract: Let b ≥ 2 be an integer. For any integer n  ≥ 0, we call `reverse' of n in base b the integer obtained by reversing the digits of n. The existence of infinitely many prime numbers whose reverse is also prime is an open problem. In this talk, we will present a joint work with Cécile Dartyge and Joël Rivat, in which we show that there are infinitely many primes with an almost prime reverse. More precisely, we show that there exist an explicit integer \Omega_b > 0 and c_b > 0 such that, for at least c_b b^ℓ / ℓ^2 primes p ∈ [b^{ℓ-1},b^ℓ[, the reverse of p has at most \Omega_b prime factors. Our proof is based on sieve methods and on obtaining a result in the spirit of the Bombieri-Vinogradov theorem concerning the distribution in arithmetic progressions of the reverse of prime numbers.

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Abstract: TBA