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 -函数的零、l -函数值的分布和相关数论对象的理解。与Zeev Rudnick合作，研究者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推测的渐近公式具有相同的数量级。在相关工作中，研究者正在开发一种获取l -函数极值的新方法，这些新想法似乎具有广泛的适用性。研究者还将致力于寻找l函数的二次扭转的某些矩的新的渐近公式。最后，与Andrew Granville一起，研究者将研究乘法数论中的问题，特别是与字符和相关的问题。这位研究者的研究领域是解析数论。数论研究的主要对象是l函数。这些是由有趣的算术对象（如质数）构建的某些解析函数。理解l函数的零点和值是现代数学的主要主题之一。除了对这些问题的内在兴趣外，该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.