Abstract. I will talk about projects in which we combine heuristics with computational data to develop a theory in problems where it was previously hard to be confident of the guesses that there are in the literature.

1/ "Speculations about the number of primes in fast growing sequences". Starting from studying the distribution of primes in sequences like $2^n-3$, Jon Grantham and I have been developing a heuristic to guess at the frequency of prime values in arbitrary linear recurrence sequences in the integers, backed by calculations.

If there is enough time I will then talk about:

2/ "The spectrum of the $k$th roots of unity for $k>2$, and beyond".  There are many questions in analytic number theory which revolve around the "spectrum", the possible mean values of multiplicative functions supported on the $k$th roots of unity. Twenty years ago Soundararajan and I determined the spectrum when $k=2$, and gave some weak partial results for $k>2$, the various complex spectra.  Kevin Church and I have been tweaking MATLAB's package on differential delay equations to help us to develop a heuristic theory of these spectra for $k>2$, allowing us to (reasonably?) guess at the answers to some of the central questions.

The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that this function often takes algebraic values that satisfy explicit reciprocity laws and that are related to derivatives of Hecke L-functions at s=0. Thus this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12^th problem. This is joint work with Nicolas Bergeron and Pierre Charollois.

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems.  This is a report on joint work with Marco Sangiovanni Vincentelli.

The main object of study of the talk is the balanced triple product p-adic L-function; this is a p-adic L-function associated with a triple of families of (quaternionic) modular forms. The first instances of these functions appear in the works of Darmon-Lauder-Rotger, Hsieh, and Greenberg-Seveso. They have proved to be effective tools in studying cases of the p-adic equivariant Birch & Swinnerton-Dyer conjecture. With this aim in mind, we discuss the construction of a new p-adic L-function, extending Hsieh's construction, and allowing classical weight one modular forms in the chosen families. Such improvement does not come for free, as it coincides with the increased dimension of certain Hecke-eigenspaces of quaternionic modular forms with non-Eichler level structure; we discuss how to deal with the problems arising in this more general setting. One of the key ingredients of the construction is a p-adic extension of the Jacquet-Langlands correspondence addressing these more general quaternionic modular forms. This is joint work in progress with Aleksander Horawa.

Many problems in number theory are equivalent to determining all of the rational points on some curve or family of curves. In general, finding all the rational points on any given curve is a challenging (even unsolved!) problem. 

The focus of this talk is rational points on so-called Erdős-Selfridge curves. A deep conjecture of Sander, still unproven in many cases, predicts all of the rational points on these curves. 

I will describe work-in-progress proving new cases of Sander's conjecture, and sketch some ideas in the proof. The core of the proof is a `mass increment argument,' which is loosely inspired by various increment arguments in additive combinatorics. The main ingredients are a mixture of combinatorial ideas and quantitative estimates in Diophantine geometry.

The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

Hoheisel used zero-density results to prove that for all x large enough there is a prime number in the interval $[x−x^{\theta}, x]$ with $θ < 1$. The connection between zero-density estimates and primes in short intervals was explicitly described in the work of Ingham in 1937. The approach of Ingham combined with the zero-density estimates of Huxley (1972) provides us with the distribution of primes in $[x−x^{\theta}, x]$ with $\theta > 7/12$. Further improvement upon the value of \theta was achieved by combining sieves with the weighted zero-density estimates in the works of Iwaniec and Jutila, Heath-Brown and Iwaniec, and Baker and Harman. The last work provides the best result achieved using zero-density estimates. We will discuss the main ideas of the paper by Baker and Harman and simplify some parts of it to show a more explicit connection between zero-density results and the sieved sums, which are used in the paper. This connection will provide a better understanding on which parts should be optimised for further improvements and on what the limits of the methods are. This project is still in progress.