A sieve problem over the Gaussian integers

Thesis

Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors.

We prove this by first developing the theory of $L$-functions for Gaussian primes by using standard methods. We then give an exposition of the Siegel--Walfisz Theorem for Gaussian primes and a corresponding Prime Number Theorem for Gaussian Arithmetic Progressions.

Finally, we prove the main result by using the developed theory together with Sieve Theory and specifically a weighted linear sieve result to bound the number of prime factors of a² + 4b². For the application of the sieve, we need to derive a specific version of the Bombieri--Vinogradov Theorem for Gaussian primes which, in turn, requires a suitable version of the Large Sieve. We are also able to get the number of prime factors of a² + 4b² as low as 3 if we assume the Generalised Riemann Hypothesis.

Authors: Schlackow, W; Dr Waldemar Schlackow

• Publication date: 2010
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: L-functions; Gaussian integers; Riemann Hypothesis; Sieve Theory; Gaussian primes
• Subjects: Mathematics; Number theory