Explicit methods in number theory

l -函数在数论中起着重要的作用，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、阿贝尔变分、数域和自同构形式。尽管l -函数具有核心作用，但它们在很大程度上仍然是神秘的。这个项目的目标是研究l函数的基本性质，特别是关于它们的值和零。我将集中精力在几个方面：黎曼ζ函数和其他l函数的矩和值分布，椭圆曲线的秩，l函数的恒等式，l函数的值和零的显式计算方法和算法。我要解决的问题包括：**1)黎曼ζ函数的值是如何分布的？甚至连黎曼ζ函数在给定高度下能达到多大这个基本问题，也没有得到很好的理解。受随机矩阵理论的启发，我相信应该有可能获得矩的完整的“一致”渐近性，然后推断出关于zeta函数的值分布的详细信息，包括它的最大大小。**2)我发现了黎曼ζ函数和狄利克雷l函数的一些有用的恒等式。这些公式遵循经典的路线，但似乎被忽略了。它们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函数是如何相互关联的。我计划探索所使用的任何方法是否可以应用于更高次的l函数，例如那些由经典模形式产生的函数。**3)我以前做了很多工作，证实了各种l函数族的矩的详细猜想。一个有趣的特征出现在ζ函数的二次扭曲中。多重狄利克雷级数理论预测了这些l函数的三次矩和高矩的额外低项。早些时候，我和我的研究生奥尔德森，为了测试这些低项，我们开发了算法。我们的结果令人鼓舞，似乎确实证实了这些说法。然而，所涉及的小常数和非常嘈杂的剩余项共同作用，使得很难提出支持额外较低项的结论性数值证据。提出了几种研究途径。**第一个是检验椭圆曲线l函数的二次扭曲的类似预测。这将提供更丰富的数据集，用于测试额外的较低项。**另一个想法是，在二次狄利克雷l函数的情况下，基于相关爱森斯坦级数的傅里叶展开，开发更快的算法，并收集更多的数据。**我还将研究函数场zeta函数，其中一个平行理论表明存在额外的低项。**4)椭圆曲线的秩可以有多大？能找到具有特殊性质的椭圆曲线吗？如何有效地计算椭圆曲线的秩？一条椭圆曲线有非平凡秩的频率是多少？这是我打算调查的一些问题。**5)我计划从计算复杂性的角度改进算法，用于计算l函数的零点和值。**我的研究将有助于数论领域，提供基础和重要的知识进步，并导致高素质人才的培训。

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.