Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)Jurkat-Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec's lower bound on Jacobsthal's problem, as well as minor improvements to the Brun-Titchmarsh Theorem. We also deduce an improved (though conditional) lower bound on the longest gaps between primes, and rework Cram\'er's heuristic in this situation to show that we would expect gaps around \( x\) that are significantly larger than \( (\log x)^2.\)

(with Regis de la Breteche)

We show that if an exponential sum with multiplicative
coefficients is large then the associated multiplicative function is
``pretentious''. This leads to applications in the circle method, and a natural interpretation of the local-global principle.

(with George Shakan and Aled Walker)

Let \( A \subset \mathbb{Z}^d\) be a finite set. It is known that \( NA\) has a particular size (\( \vert NA\vert = P_A(N)\) for some \( P_A(X) \in \mathbb{Q}[X]\) )
and structure (all of the lattice points in a cone other than certain exceptional sets), once \( N\) is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary \( A\) . Such explicit results were only previously known in the special cases when \( d=1\) ,
when the convex hull of \( A\) is a simplex or when \( \vert A\vert = d+2\) , results which we improve.