Hybrid Euler-Hadamard product for quadratic Dirichlet L-functions in function fields

Abstract

We develop a hybrid Euler‐Hadamard product model for quadratic Dirichlet L–functions over function fields (following the model introduced by Gonek, Hughes and Keating for the Riemann‐zeta function). After computing the first three twisted moments in this family of L–functions, we provide further evidence for the conjectural asymptotic formulas for the moments of the family.

The fourth moment of derivatives of Dirichlet L-functions in function fields

“…Notice the similarity between the conjecture (4) for k = 1, 2 and Conrey's result (5), and the corresponding special cases of our results:…”

“…A function field analogue of [6] can be found in the thesis of Yiasemides (yet to be published), where conjectures of all even moments of Dirichlet L-functions in function fields are given, as well as an extension of this to the first derivatives of the L-functions. With regards to moments of the family of quadratic Dirichlet L-functions in function fields, we refer the reader to the work of Andrade and Keating [1,2], and to the work of Bui and Florea [5] for an approach to conjecturing higher moments via the method of the Euler-Hadamard hybrid formula. These are but a few of the many results regarding moments of L-functions in function fields.…”

The Ratios Conjecture and upper bounds for negative moments of $L$-functions over function fields

“…In this section we shall prove Theorem 1.2. Similar results without the shifts were obtained in [8].…”

“…A 2Ω(fr) k 2Ω(fr) m 2Ω(fr ) (log 1 β ) 2Ω(fr) 2 Ω(fr) |f r | 1+2β × P |f j+1 ⇒d(P )∈I j+1 Ω(f j+1 )=s j+1 /2 , where in the last line we used Lemma 3.6 in [8] and the Prime Polynomial Theorem.…”

“…for some parameter X, where d(h) denotes the degree of the polynomial h, and we will prove asymptotic formulas for twisted, shifted moments of L-functions, generalizing the work in [8]. This will be the content of Theorem 1.2.…”