Zeros of L-functions and applications

解析数论是数论的一个分支，它通过分析和复分析来研究自然数{1，2，3，4，…}和素数{2，3，5，7，.}。素数已经被所有文明研究了数千年。素数是所有自然数的基石，因为任何自然数都是素数的乘积。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一个谜，也是数学中重要猜想的对象。在上个世纪，素数对政府和工业来说已经变得非常重要。RSA是一种在互联网交易中用于安全发送秘密消息的方法。这种方法之所以有效，是因为将一个大整数分解到它的素数因子中极其困难。解析数论是在1837年Dirichlet和1859年Riemann发表之后出现的。这两位作者指出，素数可以通过某些复函数来研究，称为L函数。一个非常特殊的L函数是黎曼Zeta函数。它可以写成形式为1+1/2^S+1/3^S+..的无穷和，其中S是复数。L-函数是类似定义的等积。狄里克莱特和黎曼都意识到L函数可以用来研究素数。Dirichlet证明了存在无穷多个具有某些特殊模式的素数，Riemann证明了它们关于Zeta函数零点的素数的存在公式。*另一个关键进展是H.L.蒙哥马利。上世纪70年代，S作了一个基本发现：黎曼Zeta函数的零点的行为类似于大型随机矩阵的特征值，他发现Riemann Zeta函数的零点的统计量本质上与一大群随机矩阵的特征值的统计量相同。他从世界顶尖物理学家之一弗里曼·戴森那里得知了这种联系。在20世纪50年代，S、戴森和其他理论物理学家一直在通过计算大型随机矩阵的特征值来模拟量子混沌中的能级。这是由Erastosthenes发明的，并由Brun、Selberg、Rosser和Bombieri进一步开发。这种方法使用组合学来研究素数，最近在检测素数之间的差距方面变得非常有效。我的研究是在狄利克莱特/黎曼、蒙哥马利和筛子方法的思想的交汇点上进行的。本文提出用显式方法研究L函数零点的统计性质。我想证明L函数的零点是简单的，Zeta函数的零点之间有很小的间隙，平均而言，L函数的零点不满足线性关系。研究L函数的零点与素数计数和及相关和的分布之间的关系。

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. ***