Primes and divisors

摘要题目：素数与除数本提案涉及与素数和整数除数的分布有关的三个项目。 第一个项目涉及继续调查（与S。Konyagin）关于算术级数中素数分布的微妙不等式，这类问题与Dirichlet L-函数和Riemann zeta函数的非平凡零点的分布密切相关。 研究的重点是进一步阐明具有不同于1/2的真实的部分的L-函数的假设零点的影响。第二个项目涉及筛子方法的局限性，这是一种限制素数数量的强大技术（或具有有限数量的素因子的数字）在一个整数序列中。提议者的工作目标是提高对序列中素数分布如何受到序列分布影响的理解算术级数和奇数数列的分布算术级数和奇数数列的分布（分别和偶数）的素因子。最后一个项目是提高在给定区间内拥有因子的整数的密度的界限，这是整数因子分布理论的核心问题。素数的研究已有2000多年的历史，and the most最important重要problems问题around how the primes素数are distributed分布. 除了应用于许多其他数学分支，素数理论是至关重要的现代编码理论和密码学，例如安全的互联网communication.The拟议的研究涉及的问题，关于如何素数分布在某些集合的整数，和问题的相关主题的分布的约数的统计行为。 建议者将涉及研究生，博士后和/或有才华的本科生（REU）在这些研究项目。

AbstractTitle: Primes and divisorsThis proposal is concerned with three projects related to the distributionof prime numbers and divisors of integers. The first project concernscontinuing investigations (jointly with S. Konyagin) of subtleinequities in the distribution of primes in arithmetic progressions.Such problems are intimately connected with the distribution of non-trivialzeros of Dirichlet L-functions and the Riemann zeta function. The researchfocus is on further illuminating the influence of hypothetical zeros of theL-functions having real part different from 1/2.A second project deals with the limitations of sieve methods, which arepowerful techniques for bounding the number of primes (or numbers witha bounded number of prime factors) in an integer sequence.The goal of the proposer's work is an improved understanding ofhow the distribution of primes in a sequence is influenced by thedistribution of the sequence in arithmetic progressions and the distributionof the sequence on numbers with an odd number (respectively and even number)of prime factors.The final project is to improve bounds for the density of integers possessinga divisor in a given interval, a problem which is central to the theory ofthe distribution of divisors of integers.The study of prime numbers is more than 2000 years old, and the most importantproblems revolve around how the primes are distributed. Besides applicationsto many other branches of mathematics, prime number theory is crucial tomodern coding theory and cryptography, e.g. secure Internet communication.The proposed research deals with questions concerning how the primes aredistributed within certain sets of integers, and questions on the relatedsubject of the statistical behavior of the distribution of divisors ofnumbers. The proposer will involve graduate students,post-docs and/or talented undergraduate students (REU) in these researchprojects.