Sieve Methods with Applications

筛子理论提供了从更大的序列中选择特别感兴趣的子序列的工具，而更大的序列通常更容易通过其他方式访问。例如，在最近的发展中，素数被捕获为四次多项式值。该方案不仅解决了素数问题，而且还解决了一些丢番图方程的求解问题和某些三次曲面上有理点的估计问题。在筛子理论的更多理论方面，该提议的目标是调查这些方法(筛子的奇偶障碍)的内在局限性，并找到打破这些局限性的方法。当用调和分析(谱方法)的论点加强筛选思想时，它成为强大而通用的工具，甚至可以超过大黎曼假说的能力。该提案在这个方向上提出了一些建议。事实证明，这几种方法对从事密码学工作的研究人员非常有吸引力。虽然这个项目没有直接解决这种应用，但筛子理论的进步似乎可能会带来新的可能性。傅立叶分析在筛分方法中的应用，对现代谐波分析产生了很大的需求，必将对塑造现代谐波分析产生有价值的影响。在组合思想和分析的界面上的这些发展将对研究生非常鼓舞人心。

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.