Number fields and function fields: Coalescences, contrasts and emerging applications

J. P. Keating, Z. Rudnick, T. D. Wooley

Research output: Contribution to journalReview articlepeer-review


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.

Original languageEnglish
Article number20140315
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2040
StatePublished - 28 Apr 2015


  • Analytic number theory
  • Arithmetic statistics
  • Exponential sums
  • Function fields
  • Zeta functions


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