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，对数字安全有着广泛的影响。这个建议主要涉及解析数论，这是这方面的主题，旨在回答定量问题。例子包括：一个典型的整数在其素因子的数量和大小方面看起来像什么？在给定的高度上有多少个素数？那么，与另一个素数相差2的素数（所谓的孪生素数）呢？这个建议考虑了一些问题，这种类型的，不仅是自然数，而且也为其他一些系统，已证明arithmeticalsignificantly（相应的号码领域和功能fields.Three具体议题被认为：第一个问题分裂统计素数在代数数域。Elliott和Linnik-Vinogradov发展了一种方法，用于在Q的阿贝尔扩张中用给定的分裂类型来界定最小有理素数。与Micah Milinovich一起，Pollack将研究将他们的思想扩展到某些非阿贝尔扩展。第二个主题是算术函数的理论，特别是当它与概率数论相联系时。要考虑的一个问题是，在经典的Erdos和Wintner定理获得误差界。最后，波拉克将继续他的研究分布的不可约多项式在有限领域，特别注意的问题，动机合理的素数理论。例如，波拉克将研究特殊配置的不可约多项式，包括类似的总理k元组猜想和贝特曼-霍恩猜想。

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.