The sieve theory offers tools for selecting subsequences of particular interest from a larger sequence which is typically more accessible by other means. For example in the recent developments prime numbers were captured in polynomial values of degree four. The proposal goes further to solve problems not only concerning prime numbers but also for solving some diophantine equations and estimating the rational points on some cubic surfaces. In more theoretical aspects of sieve theory the goal of the Proposal is to investigate the intrinsic limitations of the methods (parity barrier of sieve) and to find ways to break these limits. Sieve ideas when enhanced with arguments of harmonic analysis (spectral methods) become powerful and versatile tools which can even exceed the capability of the Grand Riemann Hypothesis. The Proposal makes a few suggestions in this direction.
Sieve methods turned out to be very attractive for researchers working in cryptography. Although this project does not address such applications directly, it seems likely that advances in the theory of sieves will open new possibilities. The implementation of Fourier analysis to sieve methods creates a lot of demand in modern harmonic analysis and will certainly have valuable impact on shaping the latter. These developments in the interface of combinatorial ideas and analysis will be quite inspiring for graduate students.
The project concerns sieve methods and their applications. Many results of this project, and many more by other investigators, are included in the book "Opera de Cribro" by J.Friedlander and the PI. The main target of sieve methods are prime numbers. However, for a long time, the sieve methods were not capable to touch primes due to the parity barrier. This barrier was crashed by Friedlander and the PI by introducing new inputs and devices to sieve theory. In the project we continue using these ideas for solving special equations with prime variables. For example we showed that certain large numbers can be represented as the sum of a square and a prime number. It turns out that sieve ideas can be also used to solve diophantine equations (for example for finding rational points on special cubic surfaces). When preparing the sieve arguments for applications we encountered difficult questions, which alone are of theoretical interest. In conjunction with these questions we established a genaral method to untangle the upper-bound sieves which occur in the process of sifting simultanuously distinct, yet correlated sequences. The modern sieve is driven by tools of harmonic analysis. Along the lines of this project one can find new ways of making use of characters and exponential sums, in particular the Kloosterman sums.
