Topics in sieve theory

Thesis

This thesis is concerned with the application of sieve theory to the study of gaps between primes. It focuses on techniques which combine sieves with methods from classical analytic number theory. We consider the following two questions: On average, how big are the squares of gaps between two consecutive primes less than x? How big is the smallest integer which appears infinitely often as a gap between m consecutive primes?

Concerning the first problem, we prove that the average size of the squares of differences between consecutive primes less than x is O(x^{0.23+ε}) for any fixed ε > 0. This improves on a result of Peck, who gave bound O(x^{0.25+ε}) in the place of O(x^{0.23+ε}). Key ingredients of this work are Harman’s sieve, Heath-Brown’s mean value theorem for sparse Dirichlet polynomials and Heath-Brown’s R* bound.

Concerning the second problem, we prove that the primes below x are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to x^{1/2+1/40−ε}. The exponent of distribution 1/2+1/40 improves on a result of Polymath, who had previously obtained the exponent 1/2+7/300. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that

liminf (p_{n+m} − p_n) = O(exp(3.8075m)).

The main new ingredient of our work is a modification of the q-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of Polymath.

Authors: Stadlmann, J

• DOI: 10.5287/ora-jnn4zn1d1
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Mathematics