The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

Abstract

For Dirichlet L-functions in Fq[T ] we obtain a hybrid Euler-Hadamard product formula. We make a splitting conjecture, namely that the 2k-th moment of the Dirichlet L-functions at 1 2 , averaged over primitive characters of modulus R, is asymptotic to (as deg R −→ ∞) the 2k-th moment of the Euler product multiplied by the 2k-th moment of the Hadamard product. We explicitly obtain the main term of the 2k-th moment of the Euler product, and we conjecture via random matrix theory the main term of the 2k-th moment of the Hadamard product. With the splitting conjecture, this directly leads to a conjecture for the 2k-th moment of Dirichlet Lfunctions. Finally, we lend support for the splitting conjecture by proving the cases k = 1, 2. This work is the function field analogue of the work of Bui and Keating. A notable difference in the function field setting is that the Euler-Hadamard product formula is exact, in that there is no error term.