Statistical problems in elementary, analytic, and algebraic number theory

数论是数学中一个非常活跃的研究领域，它与整个学科都有联系，并对数字安全产生了广泛的影响。这一建议主要涉及分析数论，这是该学科的一个方面，旨在回答定量问题。例子包括：一个典型的整数在“统计学”上看起来是什么样的，就其质因数的数量和大小而言？给定高度上有多少质数？与另一个质数相差2的质数（所谓的双质数）会怎样？这一建议考虑了许多这类问题，不仅对自然数，而且对其他已被证明具有算术意义的数系统（对应于数域和函数域）。考虑了三个特定的主题：第一个主题涉及代数数字段中素数的分裂统计。Elliott和Linnik—Vinogradov在q的阿贝尔扩展中发展了一种限定具有给定分裂类型的最小有理数素数的方法。Pollack将与Micah Milinovich一起研究将他们的思想推广到某些非阿贝尔扩展。第二个主题是算术函数理论，特别是当它与概率数论联系在一起时。要考虑的一个问题是Erdos和Wintner经典定理中的误差界的求取问题。最后，波拉克将继续他在有限域上不可约多项式分布的研究，特别关注由有理数理论引起的问题。例如，Pollack将研究不可约多项式的特殊构型，包括素数k元组猜想和贝特曼-霍恩猜想的类似物。

The theory of numbers is a very active research area in mathematics, with connections across the entire discipline and with wide-reaching consequences for digital security. This proposal principally concerns analytic number theory, which is that aspect of the subject that aims to answer quantitative questions. Examples include: What does a typical integer look like 'statistically', in terms of the number and size of its prime factors? How many primes are there up to a given height? What about primes differing from another prime by 2 (so called twin primes)? This proposal considers a number of questions of this kind, not only for natural numbers but also for other number systems that have proven arithmetically significant (corresponding to number fields and function fields).Three specific topics are considered: The first concerns splitting statistics of primes in algebraic number fields. Elliott and Linnik--Vinogradov developed a method for bounding the least rational prime with a given splitting type in an abelian extension of Q. With Micah Milinovich, Pollack will investigate extending their ideas to certain nonabelian extensions. The second topic is the theory of arithmetic functions, especially as it connects with probabilistic number theory. One problem to be considered is that of obtaining error bounds in a classical theorem of Erdos and Wintner. Finally, Pollack will continue his studies of the distribution of irreducible polynomials over finite fields, with particular attention paid to problems motivated by rational prime number theory. For instance, Pollack will study special configurations of irreducible polynomials, including analogues of the prime k-tuples conjecture and the Bateman--Horn conjecture.