Keating J P, Rudnick Z, Wooley T D
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israels.
Philos Trans A Math Phys Eng Sci. 2015 Apr 28;239(2040). doi: 10.1098/rsta.2014.0315.
The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.
素数的密度与在具有(q)个元素的有限域上定义的不可约多项式的密度之间的相似性最早是由高斯观察到的。从那时起，在数域的算术与在有限域上定义的函数域的算术之间发现了许多其他的类比。尽管至少在过去半个世纪里这是一个活跃的相互作用领域，但解析数论和函数域环境中使用的语言和技术却大不相同，这阻碍了这两个领域之间的交流。目前这种情况正在发生变化，并且通过将每个领域的思想结合在一起所激发的一些问题已经取得了实质性进展。我们在此介绍在本次西奥·墨菲会议特刊上发表的论文，其中解释了一些最新进展。
