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

• Christian Elsholtz
• 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.

## References

1. 1.
M.D. Coleman, The distribution of points at which binary quadratic forms are prime. Proc. Lond. Math. Soc. 61(3), 433–456 (1990)
2. 2.
M.D. Coleman, The distribution of points at which norm-forms are prime. J. Number Theory 41(3), 359–378 (1992)
3. 3.
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I. Math. Z. 1, 357–376 (1918)
4. 4.
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. II. Math. Z. 6, 11–51 (1920)
5. 5.
D. Hensley, An asymptotic inequality concerning primes in contours for the case of quadratic number fields. Acta Arith. 28, 69–79 (1975)
6. 6.
I.M. Kalnin$$^{{\prime}}$$, On the primes of an imaginary quadratic field which are located in sectors (Russian). Latvijas PSR Zin$$\bar{\mathrm{a}}$$ t$$\stackrel{\mathrm{n}}{,}$$ u Akadēmijas Vēstis. Fizikas un Tehnisko Zin$$\bar{\mathrm{a}}$$ t$$\stackrel{\mathrm{n}}{,}$$ u Sērija. Izvestiya Akademii Nauk Latviĭskoĭ SSR. Seriya Fizicheskikh i Tekhnicheskikh Nauk 1965(2), 83–92 (1965)Google Scholar
7. 7.
S. Knapowski, On a theorem of Hecke. J. Number Theory 1, 235–251 (1969)
8. 8.
I. Kubilyus, The distribution of Gaussian primes in sectors and contours (Russian). Leningrad. Gos. Univ. Uc. Zap. Ser. Mat. Nauk 137(19), 40–52 (1950)
9. 9.
I.P. Kubilyus, On some problems of the geometry of prime numbers (Russian). Mat. Sb. N.S. 31(73), 507–542 (1952)Google Scholar
10. 10.
OEIS Foundation Inc., The on-line encyclopedia of integer sequences (2011), http://oeis.org/
11. 11.
H. Rademacher, Primzahlen reell-quadratischer Zahlkörper in Winkelräumen. Math. Ann. 111, 209–228 (1935)
12. 12.
Z.W. Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588, version 26, 18 March 2013 (2013)Google Scholar