Topics in analytic number theory and additive combinatorics

Abstract

This thesis comprises four articles in multiplicative and additive number theory, two subfields of analytic number theory, concerning e.g. the distribution of primes, multiplicative structures and additive structures.

In the first article, we consider a combination of two breakthroughs on prime gaps (small prime gaps and large prime gaps), and improve on a previous result given by Pintz. We also apply a similar strategy to improve on previous works on lower bounds for the least prime in an arithmetic progression. The proofs rely on a variant of the Maynard-Tao theorem and arguments used in proving long prime gaps.

In the second article, we study a lower bound for the L1 norm of the exponential sum of the Möbius function over short intervals. This result extends the long interval version given by Balog and Ruzsa. The proofs are based on the Balog-Ruzsa structure and an improvement for a key lemma. In the improvement we use two different techniques-complex analysis and van der Corput’s method.

In the third article, we study Vinogradov’s three primes theorem with Piatetski-Shapiro primes. Our result significantly improves the existing results via applying the transference principle and Harman’s sieve method. Besides, we improve on a Roth-type result for Piatetski-Shapiro primes given by Merik.

In the fourth article, we study dk bounded multiplicative functions in almost all short intervals. Our results generalize the breakthrough given by Matomäki and Radziwiłł and improve on Mangerel’s result. The proofs depend on the Matomäki-Radziwiłł method and introducing restrictions on prime factors.