We prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in Fq [T ]. We consider the behaviour as deg R → ∞ and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Heath-Brown's result in the number field setting, which was subsequently improved by Soundararajan. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime.

We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $$Q \in {\mathbb {F}}_q [T]$$ Q ∈ F q [ T ] , and the asymptotic limit is as $${{\,\mathrm{deg}\,}}Q \longrightarrow \infty $$ deg Q ⟶ ∞ . This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-function.

For B ∈ F q [T ] of degree 2n ≥ 2, consider the number of ways of writing B = E 2 + γF 2 , where γ ∈ F * q is fixed, and E, F ∈ F q [T ] with deg E = n and deg F = m < n. We denote this by S γ;m (B). We obtain an exact formula for the variance of S γ;m (B) over intervals in F q [T ]. We use the method of additive characters and Hankel matrices that the author previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach; and we briefly discuss the possible extension of our result to the number of ways of writing B = E 2 + T F 2 .

We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus Q ∈ Fq[t], and the asymptotic limit is as deg Q −→ ∞. This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It also expands on the work of Conrey, Rubinstein, and Snaith who cojectured, using random matrix theory, the asymptotic main term of any even moment of the derivative of the Riemann zeta-function in the number field setting.

We prove an exact formula for the variance of the divisor function over short intervals in F p [T ], where p is a prime integer. We use exponential sums to translate the problem to one involving the ranks of Hankel matrices over finite fields. We prove several results regarding the rank and kernel structure of these matrices, and thus demonstrate their number-theoretic properties. We briefly discuss extending our method to moments higher than the second (the variance); to the k-th divisor function; to correlations of the divisor function with applications to moments of Dirichlet L-functions in function fields; and to the analogous results over F q [T ], where q is a prime power.