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

Christian Elsholtz () and Glyn Harman ()
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Christian Elsholtz: Technische Universität Graz, Institut für Mathematik A
Glyn Harman: Royal Holloway University of London, Department of Mathematics

A chapter in Analytic Number Theory, 2015, pp 65-81 from Springer

Abstract: Abstract We discuss recent conjectures of T. Ordowski and Z.W. Sun on limits of certain coordinate-wise defined functions of primes in ℚ ( − 1 ) $$\mathbb{Q}(\sqrt{-1})$$ and ℚ ( − 3 ) $$\mathbb{Q}(\sqrt{-3})$$ . Let p ≡ 1 mod 4 $$p \equiv 1\bmod 4$$ be a prime and let p = a p 2 + b p 2 $$p = a_{p}^{2} + b_{p}^{2}$$ be the unique representation with positive integers a p > b p $$a_{p} > b_{p}$$ . Then the following holds: lim N → ∞ ∑ p ≤ N , p ≡ 1 mod 4 a p k ∑ p ≤ N , p ≡ 1 mod 4 b p k = ∫ 0 π ∕ 4 cos k ( x ) d x ∫ 0 π ∕ 4 sin k ( x ) d x . $$\displaystyle{\lim _{N\rightarrow \infty }\frac{\sum _{p\leq N,p\equiv 1\bmod 4}a_{p}^{k}} {\sum _{p\leq N,p\equiv 1\bmod 4}b_{p}^{k}} = \frac{\int _{0}^{\pi /4}\cos ^{k}(x)\,dx} {\int _{0}^{\pi /4}\sin ^{k}(x)\,dx}.}$$ 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.

Keywords: Primary:; 11N05 Distribution of primes; Secondary:; 11A41 primes; 11R44 Distribution of prime ideals; 11E25 Sums of squares and representations by other particular quadratic forms (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-22240-0_4

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