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

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.

On the occasion of Helmut Maier’s 60th birthday. With admiration for his beautiful results on the distribution of primes