Topics in analytic number theory and automorphic forms

Thesis

This thesis concerns improved results in the analytic theory of automorphic forms, as well as their applications to classical problems about the primes and related arithmetic objects.

First, we prove new large sieve inequalities for the Fourier coefficients of exceptional Maass forms of a given level, weighted by sequences with sparse Fourier transforms. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of n²+1 is infinitely often greater than n^1.3, improving Merikoski's previous threshold of n^1.279.

We combine these results with other ideas to show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to x^{5/8 - o(1)}, using triply-well-factorable weights for the primes. This completely eliminates the dependency on Selberg's eigenvalue conjecture in previous works of Lichtman and the author, which built in turn on works of Maynard and Drappeau. As applications, we prove refined upper bounds for the counts of twin primes and consecutive smooth numbers.

Next, we obtain density theorems for ‘exceptional’ cuspidal automorphic representations of GL_n, which fail the generalized Ramanujan conjecture at some place. We depart from approaches based on Kuznetsov-type trace formulae, and instead rely on Rankin-Selberg L-functions. This improves previous density results near the threshold of the pointwise bounds.

Building on these ideas, we develop a new approach to large sieve inequalities for families of automorphic L-functions L(s), improving earlier results and simultaneously handling the Dirichlet coefficients of L, L^-1, and log L. Our bounds are sharp in ranges that are complementary to large sieve inequalities based on trace formulae. We apply our results to establish zero density estimates for families of automorphic L-functions.

Authors: Pascadi, A

• DOI: 10.5287/ora-bpodepgeq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: automorphic forms; distribution of primes; Kloosterman sums; generalized Ramanujan conjecture; Selberg eigenvalue conjecture
• Subjects: Number theory; Automorphic forms