A wide variety of problems of multiplicative number theory will be pursued. The problem of gaps between sums of two squares will be tackled in a new way, which if successful could be developed into a major new tool for the study of the coefficients of automorphic functions in short intervals. Sieve methods will be studied not by {it ad hoc} choices of sifting functions but rather by allowing the sieve to reveal where its extremal configurations lie. The goal is to determine, in all dimensions, the optimal upper and lower bounds. The local distribution of zeros of the zeta function will be studied by locating optimal kernels to use in conjunction with pair correlation information. Statistics relating to the distribution of primes in arithmetic progressions as one averages over different arithmetic progressions will be determined.
For several decades, Hardy and Littlewood maintained a list of research problems. Among the problems remaining on their final list is to derive a better upper bound for the gap between numbers that can be expressed as a sum of two squares. Here better means simply better than the trivial bound one obtains by the greedy algorithm. This problem is to be attacked in a new way, and if the approach is successful, then the method may apply in greater generality. As was pointed out by Selberg, the problem of sieving efficiently is fundamentally a problem of linear programming. Existing sieves are somewhat ad hoc, and in most cases the bound obtained is either not optimal or at least not known to be optimal. It is proposed to emphasize the linear programming aspect of the problem, and to identify extremals, for both the primal and the dual problems. This is to be done first in the simplest of situations, and then in successively more challenging ones, so that eventually one will be able to identify extremals in realistic problems of great interest.
