Primes and divisors

摘要标题：素数和除数本提案涉及三个与整数的素数和除数的分配有关的项目。第一个项目是关于算术级数中素数分布的微妙不等价性的持续研究(与S.Kanyagin合作)。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实部不同于1/2的l-函数的假设零点的影响。第二个项目涉及筛选方法的局限性，该方法是一种强大的技术，用于限定整数序列中的素数(或具有有限数量的素因数的数)。作者的工作的目的是更好地理解序列中素数的分布如何受序列在算术级数中的分布以及序列在具有奇数和偶数的数上的分布的影响。最后一个项目是改善在给定区间内具有因子的整数的密度的界，一个关于整数的因子分布理论的核心问题。素数的研究已有2000多年的历史，最重要的问题围绕着素数如何分布展开。除了在许多其他数学分支中的应用，素数理论在现代编码理论和密码学中也是至关重要的，例如安全的互联网通信。建议的研究涉及如何在特定的整数集合内分布素数的问题，以及与数的因数分布的统计行为相关的问题。提倡者将让研究生、博士后和/或有才华的本科生(REU)参与这些研究项目。

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.