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On Conjectures of T. Ordowski and Z.W. Sun Concerning Primes and Quadratic Forms

  • Christian ElsholtzEmail author
  • Glyn Harman
Chapter

Abstract

We discuss recent conjectures of T. Ordowski and Z.W. Sun on limits of certain coordinate-wise defined functions of primes in \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\). Let \(p \equiv 1\bmod 4\) be a prime and let \(p = a_{p}^{2} + b_{p}^{2}\) be the unique representation with positive integers \(a_{p} > b_{p}\). Then the following holds:
$$\displaystyle{\lim _{N\rightarrow \infty }\frac{\sum _{p\leq N,p\equiv 1\bmod 4}a_{p}^{k}} {\sum _{p\leq N,p\equiv 1\bmod 4}b_{p}^{k}} = \frac{\int _{0}^{\pi /4}\cos ^{k}(x)\,dx} {\int _{0}^{\pi /4}\sin ^{k}(x)\,dx}.}$$
For k = 1 this proves, but for k = 2 this disproves the conjectures in question. We shall also generalise the result to cover all positive definite, primitive, binary quadratic forms. In addition we will discuss the case of indefinite forms and prove a result that covers many cases in this instance.

2010 Mathematics Subject Classification

Primary: 11N05 Distribution of primes Secondary: 11A41 primes 11R44 Distribution of prime ideals 11E25 Sums of squares and representations by other particular quadratic forms 

Notes

Acknowledgements

The first named author was partially supported by the Austrian Science Fund (FWF): W1230.

The authors thank the referee for helpful suggestions.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Mathematik ATechnische Universität GrazGrazAustria
  2. 2.Department of MathematicsRoyal Holloway University of LondonEghamUK

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