Narrow Results

We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We obtain an asymptotic formula of the fourth moment for the central value of these Dirichlet L-functions. In addition, we find a lower bound for the 2kth moment of these L-functions. These results agree up to constants with the polynomial ring analog of the Keating and Snaith Conjecture for the asymptotic of leading terms.

We study the natural analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture to have a larger constant, then this modified conjecture is satisfied by a positive proportion of hyperelliptic curves.

Let ${𝔽}_q[t]$ be the polynomial ring over the finite field ${𝔽}_q$ of $q$ elements. For a natural number $N,$ let ${𝔾}_N$ be the set of all polynomials in ${𝔽}_q[t]$ of degree less than $N.$ Let $h$ be a quadratic polynomial over ${𝔽}_q[t].$ Suppose that $h$ is intersective, that is, which satisfies $(A-A)\cap (h({𝔽}_q[t])\setminus \{0\})\ne \varnothing$ for any $A\subseteq {𝔽}_q[t]$ with ${limsup}_{N\to \infty }|A\cap {𝔾}_N|/q^N>0,$ where $A-A$ denotes the difference set of $A.$ Let $B\subseteq {𝔾}_N.$ Suppose that $(B-B)\cap \left(h({𝔽}_q[t])\setminus \{0\}\right)=\varnothing$ and that the characteristic of ${𝔽}_q$ is not divisible by 2. It is proved that $\left|B\right|\le C{N}^{-clogloglogN}{q}^N$ for any $0<c<1/log3,$ where $C\ge 1$ is a constant depending only on $q,h$ and $c.$