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，涉及整个学科，对数字安全具有广泛的影响。这项建议主要涉及解析数论，这是这门学科的一个方面，旨在回答量化问题。例如：一个典型的整数在统计上是什么样子的，就其素数的数量和大小而言？在给定的高度上有多少个素数？如果素数与另一个素数相差2(所谓的孪生素数)呢？该方案考虑了许多此类问题，不仅针对自然数，还针对已证明具有算术意义的其他数系统(对应于数域和函数域)。考虑了三个具体主题：第一个涉及代数数域中素数的分裂统计。Elliott和Linnik-Vinogradov发展了一种方法，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。与米卡·米利诺维奇一起，波拉克将研究将他们的想法推广到某些非阿贝尔扩张。第二个主题是算术函数论，特别是当它与概率数论联系在一起时。一个要考虑的问题是在Erdos和Wintner的经典定理中获得误差界的问题。最后，波拉克将继续研究有限域上不可约多项式的分布，特别关注由有理素数理论引发的问题。例如，波拉克将研究不可约多项式的特殊构型，包括质数k-字节组猜想和贝特曼-霍恩猜想的类似。

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.