1. On some problems involving Dirichlet L-functions
- Author
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Smith, Kevin, Andrade, Julio, and Byott, Nigel
- Abstract
In this thesis we study three problems in analytic number theory. These problems are related at a fundamental level, but a general introduction to those relationships would digress from the subsequent content and a summary would certainly be superficial. As such, each chapter includes a self-contained introduction and statement of results so that it may be read independently, but references to more unified literature are given where appropriate. Chapter 1 addresses the problem of computing asymptotic formulae for the expected values and second moments of central values of primitive Dirichlet L-functions L(s,χ₈d ⊗ ψ) when ψ is a fixed even primitive non-quadratic character of odd modulus q, χ₈d is a primitive quadratic character and d ≡ h (mod r) is odd and squarefree, h is fixed and r ≡ 0 (mod q) is even. Restricting to these arithmetic progressions ensures that such sets of L-functions form a "family of primitive L-functions" in the specific sense defined by Conrey, Farmer, Keating, Rubinstein and Snaith. Soundararajan had previously computed these statistics without restricting to such arithmetic progressions. We find that this restriction introduces non-negligible non-diagonal terms to the second moments that require significantly more detailed analysis to handle. Chapter 2 focuses on the moments and subconvexity of the Riemann zeta function ζ(σ+it) in the right-half of the critical strip 1/2 < σ < 1 from a functional-analytic perspective. We examine a correspondence between the moments and the Hilbert space B² of Besicovitch almost-periodic functions and a certain subgroup U of its unitary transformation group. We define a family of Hilbert spaces which are closely related to subconvexity that contain B² as a dense subset and, as a consequence of the continuity on those spaces of the transformations in U, we give a conditional proof of the conjectured asymptotic formula for the sixth moment for every fixed 1/2 < σ < 1 which, in turn, implies new bounds for the sixth moment on the critical line. We also show that the Lindelöf hypothesis is a consequence of the continuity of certain more general multiplication operators on those spaces. We conclude with the corollary that the Lindelöf hypothesis implies that a recent conjecture of Gonek, Hughes and Keating holds in the right-half of the critical strip. In Chapter 3 we consider the general additive divisor problem. Here the divisor functions d_k(n) are the number of ways of writing a natural number n as a product of k factors, and the problem is that of establishing asymptotic formulae for the correlations ∑ₙ≤ₓd_k(n)d_ℓ(n+h) with h,k,ℓ∈N. We show that the conjectured asymptotic formulae hold when one or both of the divisor functions are replaced by the minorants d_k(n,A)=∑_m|n,m≤n^Ad_k-1(m) with A sufficiently small, leading us to obtain new lower bounds for the asymptotics in the original problem. The main arguments rest on a study of the distribution of the functions d_k(n,A) in arithmetic progressions.
- Published
- 2023