Problems in Analytic Number Theory

将探讨乘法数论的各种各样的问题。用一种新的方法来解决两个平方和之间的间隙问题，如果成功的话，可以发展成为研究短时间内自同构函数系数的一个重要的新工具。筛选方法的研究将不是通过筛选函数的特别选择，而是通过允许筛选显示其极端配置所在的位置。目标是在所有维度中确定最优上界和下界。zeta函数零点的局部分布将通过定位与对相关信息结合使用的最优核来研究。将确定与等差数列中素数分布有关的统计数据，即不同等差数列的一个平均值。几十年来，哈代和利特尔伍德一直在研究一系列问题。在他们的最终列表中，剩下的问题之一是推导出可以用两个平方和表示的数字之间的差的更好的上界。这里更好的意思是比贪婪算法得到的平凡界更好。要用一种新的方法来解决这个问题，如果这种方法是成功的，那么这种方法就可以应用于更广泛的领域。正如Selberg所指出的，有效筛分的问题本质上是一个线性规划问题。现有的筛分有些特别，在大多数情况下，得到的边界不是最优的，或者至少不知道是最优的。建议强调问题的线性规划方面，并对原始问题和对偶问题进行极值识别。首先要在最简单的情况下做到这一点，然后依次在更有挑战性的情况下做到这一点，这样，最终人们将能够在非常有趣的现实问题中识别极端情况。

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.