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Principal forms X 2+nY 2 representing many integers

  • David Brink
  • Pieter Moree
  • Robert OsburnEmail author
Article

Abstract

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X 2+nY 2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.

Keywords

Binary quadratic forms Bernays’ constant Special values of L-series 

Mathematics Subject Classification (2000)

11E16 11M20 

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Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity College DublinDublin 4Ireland
  2. 2.Max-Planck-Institut für MathematikBonnGermany

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