Topics in the Analytic Theory of $L$-functions

研究者和他的同事们研究了L-函数解析理论中的几个问题。 这些项目的长期目标是提高对L-函数零点的理解，以及对L-函数值的分布和相关数论对象的理解。 与Zeev Rudnick合作，研究人员正在开发方法来获得许多L-函数族的下界。 这些下界与Keating和Snaith的渐近公式具有相同的数量级。 在相关的工作中，研究者正在开发一种新的方法来获得极值的L-函数，新兴的思想似乎是广泛的适用性。研究者还将致力于寻找新的渐近公式的某些时刻的二次扭曲的L-函数。最后，研究者将与安德鲁Granville，从乘法数论的问题，特别是有关的特征和。 该研究员的研究领域是解析数论。 数论的中心研究对象是L-函数。 这些是由算术中有趣的对象（如素数）构建的某些解析函数。 了解L-函数的零点和值是现代数学的重要课题之一。 除了对这些问题的内在兴趣之外，这一领域的进展对密码学和编码理论具有实际意义。

The investigator and his colleagues study several problems in the analytic theory of L-functions. The long term goal of these projects is to gain an improved understanding on zeros of L-functions, and on the distribution of values of L-functions and related number theoretic objects. Incollaboration with Zeev Rudnick, the investigator is developing methodsto obtain lower bounds in many families of L-functions. These lower bounds are of the same order of magnitude as the conjecturedasymptotic formulae of Keating and Snaith. In related work, theinvestigator is developing a new method for obtaining extreme values ofL-functions, and the emerging ideas seem to be of wide applicability.The investigator will also work on finding new asymptotic formulaefor certain moments of quadratic twists of L-functions.Lastly, with Andrew Granville, the investigator will work onquestions from multilplicative number theory, especiallyrelated to character sums. The investigator's area of research is analytic number theory. Thecentral objects of study in number theory are L-functions. Theseare certain analytic functions built out of arithmetically interesting objects such as prime numbers. Understanding the zeros and values of L-functions is one of the major themes of modern mathematics. Apart from the intrinsic interest in these questions, progress in this area has practical implications for cryptography and coding theory.