PhD Studentship: Arithmetic Moments and Random Matrix Statistics over Function Fields

The study of L-functions — their moments, zero statistics, and connections to random matrix theory — sits at the heart of modern analytic number theory. Function fields over finite fields 𝔽_q provide a setting where deep conjectures first posed for classical L-functions become rigorously provable, thanks to tools unavailable over the integers, including the Riemann Hypothesis for curves over finite fields (Weil) and the equidistribution results of Katz and Sarnak. This project develops three interlinked strands of research at the interface of number theory, arithmetic statistics, and random matrix theory. Strand 1: Moments of L-functions at s = 1. For a family of quadratic Dirichlet characters χ_D associated to squarefree polynomials D ∈ 𝔽_q[T], the analytic class number formula relates the special value L(1,χ_D) directly to the class number h(D) of the corresponding function field. While moments of L(1/2,χ_D) at the central point are well studied, moments of the shape M_k(q) = Σ_{deg D = 2g+1} L(1,χ_D)^k remain comparatively unexplored.

This strand develops asymptotic formulae for M_k(q) as q or the degree of D grows, with direct consequences for the statistical distribution of class numbers h(D) and for related counting problems in arithmetic statistics. Analogous questions for other families of L-functions may also be pursued. Strand 2: Correlations of arithmetic function over 𝔽_q[T]. Building on algebraic approaches that encode divisor-type functions via the cycle structure of permutations associated to polynomials over 𝔽_q[T], this strand studies shifted correlation sums Σ_{deg f = n} d_k(f) d_k(f+a) for the k fold divisor function d_k and fixed shift a ∈ 𝔽_q[T], together with their connections to the moments in Strand 1 via multiplicative structure and to conjectural analogues of the classical divisor-correlation (Ingham/Titchmarsh-type) problems. Other problems concerning the distribution of arithmetic functions over function fields may also be investigated. Strand 3: Moments of moments of characteristic polynomials. For a random matrix A drawn from the unitary, symplectic, or orthogonal groups with characteristic polynomial Λ_A(θ), the moments of moments M_N(k,β) = 𝔼[ ( (1/2π) ∫₀^{2π} |Λ_A(θ)|^{2β} dθ )^k ] encode fine statistical information connecting random matrix theory to Painlevé equations and partition functions in mathematical physics.

Extending recent results on joint moments of derivatives of Λ_A across matrix ensembles, this strand develops new formulae for M_N(k,β) in previously untreated Engineering and Physical Sciences Research Council Doctoral Landscape Award regimes of k and β, strengthening the emerging dictionary between these statistics and moments of L-functions established in Strand 1. Across all three strands, the project offers scope to explore AI-assisted methods — symbolic computation and machine-learning-based pattern detection — as a complementary tool for conjecture generation and numerical verification alongside classical analytic and algebraic techniques, providing interdisciplinary experience at the interface of pure mathematics and computation. Scope and flexibility. The three strands above are intended as guidance for the general direction of the research rather than a fixed programme. Other projects, problems and topics within analytic number theory, function-field arithmetic and random matrix theory may be selected by the supervisors and pursued instead, depending on the background and interests of the successful candidate and in agreement with the supervisors. The successful candidate will be trained in at least one of the following: analytic number theory, function-field arithmetic, and random matrix theory.

Funded by: EPSRC Engineering and Physical Sciences Research Council 

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