Explicit methods in number theory

L函数在数论中起着基础性的作用，它编码了关于各种重要算术对象的深层信息，如素数、椭圆曲线、阿贝尔簇、数域和自同构形式。尽管L函数扮演着核心角色，但它们在很大程度上仍然是个谜。**本项目的目标是研究L函数的基本性质，特别是关于它们的值和零。我将集中讨论几个方面：Riemann Zeta函数和其他L函数的矩和值分布，椭圆曲线的秩数，L函数的恒等式，计算L函数的值和零点的显式方法和算法。我将解决的问题包括：**1)Riemann Zeta函数的值是如何分布的？甚至连Riemann Zeta函数可以达到多大的基本问题，直到给定的高度，也没有得到很好的理解。在随机矩阵理论的启发下，我相信应该可以得到矩的完全一致渐近性，然后推导出关于Zeta函数的值分布的详细信息，包括它的最大尺寸。**2)我发现了Riemann Zeta函数和Dirichlet L-函数的一些有用的恒等式。这些公式都是经典的，但似乎被遗漏了。它们是基于我愉快地探索过的求和方法。这些恒等式可以用来从解析的角度研究这些L函数，也可以用于高精度的数值计算。这些公式还显示了不同的L函数是如何相互联系的。我计划探索所用的方法中是否有任何方法可以应用于高次L函数，例如由经典模形式产生的那些方法。**3)我以前做了大量的工作来证实关于各种L函数族的矩的详细猜想。Zeta函数的二次扭曲出现了一个有趣的特征。多重Dirichlet级数理论预测了这些L函数的三次和高阶矩的额外低阶项。早些时候，我们和我的研究生奥尔德森一起开发了算法，以测试这些较低的术语。我们的结果令人鼓舞，似乎确实证实了这些条款。然而，所涉及的小常量和非常嘈杂的余项合在一起，使得很难提出支持额外较低项的确凿数字证据。**提出了几个研究途径。**第一个是检验类似的关于椭圆曲线L函数的二次扭度的预测。这将提供一个更丰富的数据集，用来测试超低项的存在。**对于二次狄里克莱特L函数，另一个想法是基于相关爱森斯坦级数的傅里叶展开，开发更快的算法，并收集更多数据。**我还将研究函数域Zeta函数，其中并行理论表明存在超低项。**4)椭圆曲线的秩能达到多大？人们能找到具有不寻常性质的椭圆曲线吗？计算椭圆曲线的阶数的效率有多高？一条椭圆曲线多久有一次非平凡的秩次？这些是我计划调查的一些问题。**5)我计划从计算复杂性的角度改进计算L函数的零点和值的算法。**我的研究将为数论领域做出贡献，提供基础和重要的知识进步，并培养高素质的人才。

L-functions play a fundamental role in number theory and encode deep information concerning a wide variety of important arithmetic objects such as prime numbers, elliptic curves, abelian varieties, number fields, and automorphic forms. In spite of their central role, L-functions remain largely mysterious.**The goal of this project is to investigate fundamental properties of L-functions, specifically concerning their values and zeros. I will focus my efforts on several areas: the moments and value distribution of the Riemann zeta function and other L-functions, ranks of elliptic curves, identities for L-functions, explicit methods and algorithms for computing the values and zeros of L-functions. The problems that I will address will include:**1) How are the values of the Riemann zeta function distributed? Even the basic question of how large the Riemann zeta function can get, up to given height, is not well understood. Motivated by insights provided by random matrix theory, I believe it should be possible to obtain the full *uniform* asymptotics of the moments and then deduce detailed information regarding the value distribution of the zeta function, including its maximal size.**2) I have discovered a number of useful identities for the Riemann zeta function and Dirichlet L-functions. These formulas are along classical lines but seem to have been missed. They are based on summation methods that I have joyfully explored. The identities can be used to study these L-functions from an analytic point of view, and also for purposes of high precision numerical computation. The formulas also show how different L-functions interrelate. I plan to explore whether any of the methods used can be applied to higher degree L-functions, such as those arising from classical modular forms.**3) I have previously done much work confirming detailed conjectures for the moments of various families of L-functions. An interesting feature occurs for quadratic twists of the zeta function. The theory of multiple Dirichlet series predicts extra lower terms for the cubic and higher moments of these L-functions. Earlier, with my grad student Alderson, we developed algorithms in order to test for these lower terms. Our results were encouraging and do seem to confirm such terms. However, the small constants involved and very noisy remainder term conspire to make it hard to claim conclusive numerical evidence in favour of the extra lower terms. **Several avenues of research are proposed.**The first is to examine a similar prediction for quadratic twists of elliptic curve L-functions. This will provide a richer data set with which to test for extra lower terms.**Another idea, in the case of quadratic Dirichlet L-functions, would be to develop faster algorithms, based on the Fourier expansion of related Eisenstein series, and gather more data.**I will also examine function field zeta functions, where a parallel theory suggests the existence of extra lower terms.**4) How large can the rank of an elliptic curve get? Can one find elliptic curves with unusual properties? How efficiently can one compute the rank of an elliptic curve? How often does an elliptic curve have non-trivial rank? These are some of the questions that I plan to investigate. **5) I plan to work on improving algorithms, from the point of view of computational complexity, for computing zeros and values of L-functions.**My research will contribute to the field of number theory, providing fundamental and important advances in knowledge, and also resulting in the training of highly qualified personnel.