Aled Walker

Generating Z/qZ using primes

In the late 1980s Erdos asked, for large primes q, whether every non-zero residue class modulo q can be expressed as the product of two primes, each less than q. Although this conjecture is still open, even assuming the extended Riemann hypothesis, we will present a bouquet of past and present results which together constitute some partial progress towards it. These will include a positive density theorem for products of two primes, a result for almost-primes, and results taken by averaging over q. The methods will be extremely varied in nature, combining classical additive-combinatorial results on sets of small doubling with other techniques from Fourier analysis and sieve theory. No prior knowledge of analytic number theory will be required or assumed.