In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers A and B have natural density 1, then their product set A · B := {ab : a ϵ A, b ϵ B also has natural density 1. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set A of density 1 such that A · A has a “large” complement.

Drawing inspiration from the work of Nathanson and Yamada, we prove an effective and explicit version of Chen's theorem. By contrast, existing proofs of Chen's theorem are ineffective due to their use of the Siegel-Walfisz theorem. Our main result is that every even integer larger than exp(exp(32.7)) can be written as the sum of a prime and the product of at most two primes. We also prove that all even integers N >= 4 can be written as the sum of a prime and the product of at most e(29.3) primes. The main idea will be to follow a proof of Chen's theorem due to Nathanson, being more careful with the treatment of potential Siegel zeros in order to obtain an effective and explicit result. In following this framework we also prove an explicit version of the linear sieve, which substantially improves upon the previous best one by Nathanson.

We generalise a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefficients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial congruences and polynomial values which generalises a result of Hooley. We also obtain some new results for mixed character sums.