Theory of L-functions, prime numbers and divisors

点击翻译按钮获取中文摘要

This project is concerned with investigations into the distributionof zeros of L-functions, prime numbers and divisors of integers.The distribution of prime numbers and zeros of L-functions have longbeen known to be related, but some relations have only recently cometo light. Together with A. Zaharescu and K. Soundararajan, theinvestigator is exploring connections between fractionalparts of imaginary parts of zeros of L-functions, counts ofprime numbers in short intervals, and the pair correlation of zerosof L-functions. A second area of study involves uncoveringsubtle inequities in the distribution of numbers with exactlyk prime factors (k1) in arithmetic progressions, problems intimatelyconnected with the location of the zeros of Dirichlet L-functions.A focus of the research is examining differences between thethe behavior in arithmetic progressions of primes (k=1) and "almostprimes" (k1). Thirdly, the investigator will continue studyinghow the divisors of "typical" and "atypical" integers are distributed.Investigations into the probability theory underlying the study ofdivisors will be a major theme.The positive integers are perhaps the most basic objects inmathematics, and it is important to understand their multiplicativestructure - how they may be written as products, both ofprime numbers and in general. Questions about the distribution ofintegers with a certain multiplicative structure, especially primenumbers, are often very difficult and poorly understood.Several problems of this type are being investigated in this project, withparticular emphasis on the connections with the behavior ofspecial functions called L-functions and with probability andstatistical theory.