Zeros of L-functions and applications

解析数论是研究自然数{1,2,3,4，…}和质数{2,3,5,7，。通过分析和复分析。质数已经被所有文明研究了数千年。素数是所有自然数的基础，因为任何自然数都可以分解成素数的乘积。例如，100=2*2*5*5。尽管它们的定义很简单，但它们在其他整数中的出现仍然是神秘的，并且是数学中重要猜想的对象。在上个世纪，质数对政府和工业来说变得非常重要。RSA是一种在互联网交易中用于安全地发送秘密信息的方法。这种方法之所以有效，是因为将一个大整数分解成它的质因数是极其困难的。解析数论产生于1837年狄利克雷和1859年黎曼的论文发表之后。这些作者表明，素数可以通过某些称为l函数的复函数来研究。一个非常特殊的l函数是黎曼函数。它可以写成1+1/2^s+1/3^s+。其中s是复数。l函数是一个类似定义的等和。狄利克雷和黎曼都意识到l函数可以用来研究素数。狄利克雷证明了有无限多个素数具有一定的特殊模式，黎曼则证明了存在以零点表示的素数公式。***另一项关键的进展要归功于h·l·蒙哥马利。在20世纪70年代，他做出了根本性的发现黎曼ζ函数的零点表现得像大型随机矩阵的特征值。他发现黎曼ζ函数的零点统计本质上与一大组随机矩阵的特征值统计相同。他从世界领先的物理学家弗里曼·戴森那里得知了这种联系。在20世纪50年代，戴森和其他理论物理学家一直在用大随机矩阵的特征值来模拟量子混沌中的能级。第三个中心思想是组合/筛选方法。这是由埃拉斯托斯特尼发明的，并由布朗、塞尔伯格、罗塞尔和Bombieri进一步发展。这种方法使用组合学来研究素数，最近在检测素数之间的间隙方面变得非常有效。****我的研究是Dirichlet/Riemann， Montgomery和筛法思想的结合。我建议用显式公式方法研究l函数零点的统计性质。我想说明的是l函数有简单的零点，函数的零点之间有小的间隙，平均来说，l函数的零点不满足线性关系。我想研究l函数的零点与素数计数和及相关和的分布之间的联系。* * * * * * * *

Analytic number theory is the branch of number theory that studies the natural numbers {1,2,3,4,...} and the prime numbers {2,3,5,7,.} via analysis and complex analysis. Prime numbers have been studied for thousands of years, by all civilizations. Prime numbers are the building blocks of all natural numbers, since any natural number factors into a product  of primes. For instance, 100=2*2*5*5. Despite their simple definition, their occurrence among the other integers remains mysterious and the object of important conjectures in mathematics.  In the last century, prime numbers have become very important for governments and industry. RSA is a method used in internet transactions for sending secret messages securely. This method works because it is extremely difficult to factor a large integer into its prime factors. Analytic number theory arose after publications of Dirichlet in 1837 and Riemann in 1859. These authors showed that prime numbers can be studied via certain complex functions, called L-functions. A very special L-function is the Riemann zeta function. It may be written as an infinite sum of the form 1+1/2^s+1/3^s+..where s is a complex number. An L-function is an similarly defined isum. Both Dirichlet and Riemann realized that L-functions could be used to study prime numbers. Dirichlet showed that there are infinitely many primes having certain special patterns and Riemann showed that their exist formulae for primes in terms of zeros of the zeta function.***Another key advance is due to H. L. Montgomery. In the 1970's he made the fundamental discovery that the zeros of the Riemann zeta function behave like the eigenvalues of large random matrices. He discovered that the statistics of the zeros of the Riemann zeta function are essentially the same as the statistics of the eigenvalues of a large collection of random matrices. He learned of this link from Freeman Dyson, one of the world's leading physicists. In the 1950's, Dyson and other theoretical physicists had been modelling energy levels in quantum chaos by the eigenvalues of large random matrices. ***A third central idea is the combinatorial/sieve method.  This was invented by Erastosthenes and further developed by Brun, Selberg, Rosser, and Bombieri. This methods uses combinatorics to study prime numbers and recently has become very efficient in detecting gaps between primes.  ****My research lies at the interface of the ideas of Dirichlet/Riemann, Montgomery and the sieve method. I propose to study statistical properties of the zeros of L-functions via explicit formula methods. I wish to show there are simple zeros of L-functions, that there are small gaps between zeros of the zeta function, and that on average the zeros of L-functions do not satisfy linear relations. I wish to investigate the connection between zeros of L-functions and the distribution of primes counting sums and related sums. ********