Problems in Analytic Number Theory

点击翻译按钮获取中文摘要

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 ofthe coefficients of automorphic functions in short intervals. Sieve methodswill be studied not by {\it ad hoc} choices of sifting functions but rather byallowing the sieve to reveal where its extremal configurations lie. The goal isto determine, in all dimensions, the optimal upper and lower bounds. The localdistribution of zeros of the zeta function will be studied by locating optimalkernels to use in conjunction with pair correlation information. Statistics relating to the distribution of primes in arithmetic progressions asone averages over different arithmetic progressions will be determined. For several decades, Hardy and Littlewood maintained a list of researchproblems. Among the problems remaining on their final list is toderive a better upper bound for the gap between numbers that can be expressedas a sum of two squares. Here better means simply better than the trivialbound one obtains by the greedy algorithm. This problem is to be attackedin a new way, and if the approach is successful, then the method may applyin greater generality. As was pointed out by Selberg, the problem of sieving efficiently isfundamentally a problem of linear programming. Existing sieves aresomewhat ad hoc, and in most cases the bound obtained is either notoptimal or at least not known to be optimal. It is proposed toemphasize the linear programming aspect of the problem, and to identifyextremals, for both the primal and the dual problems. This is to bedone first in the simplest of situations, and then in successively morechallenging ones, so that eventually one will be able to identifyextremals in realistic problems of great interest.