arXiv subjects

J. MacMillan

Publications and source records attributed to J. MacMillan.

5 recordsLinked to original sources

arXiv2023

A Note on Twisted Moments of Dirichlet $L$-functions

In this paper, we establish an asymptotic formula for the twisted second moments of Dirichlet $L$-functions with one and two twists when averaged over all primitive Dirichlet characters of modulus $R$, where $R$ is a monic polynomial in $\mathbb{F}_q[T]$. The main result in this paper generalizes the work of Djanković [`The reciprocity law for the twisted second moment of Dirichlet $L$-functions over rational function fields', Bull. Aust. Math. Soc. 98 (2018), no. 3, 382--388].

math.NT↗

arXiv2021

Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic

In this paper, we extend to the function field setting the heuristics developed by Conrey, Farmer, Keating, Rubinstein and Snaith for the integral moments of L-functions. Also, we adapt to function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, we obtain an asymptotic formula for the integral moments and ratios of the quadratic Dirichlet L-functions $L(s,χ_u)$ over the rational function field $\mathbb{F}_q(T)$, when $q$ is a power of 2 and over a given family. As an application, we calculate the one-level density for the zeros of these L-functions.

math.NT↗

arXiv2020

Rudnick and Soundarajan's Theorem over Prime Polynomials for the Rational Function Field

In this paper, we use the methods of Andrade, Rudnick and Soundarajan to prove a Theorem about Lower bounds of moments of quadratic Dirichlet L-functions associated to monic irreducible polynomials over function fields.

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arXiv2019

Remark on a Simple Proof of the Mean Value of $K_2(\mathcal{O})$ in Function Fields

Let $\mathbb{F}_q$ denote a finite field of odd cardinality $q$, $\mathbb{A}=\mathbb{F}_q[T]$ the polynomial ring over $\mathbb{F}_q$ and $k=\mathbb{F}_q(T)$ the rational function field over $\mathbb{F}_q$. In this paper, we compute the average value of the size of the group $K_2(\mathcal{O}_{γD})$, where $\mathcal{O}_{γD}$ denotes the integral closure of $\mathbb{A}$ in $k(\sqrt{γD})$, $D$ is a monic, square-free polynomial of even degree and $γ$ is a fixed generator of $\mathbb{F}_q^*$.

math.NT↗

arXiv2019

The First Moment of $L(\frac{1}{2},χ)$ for Real Quadratic Function Fields

In this paper we use techniques first introduced by Florea to improve the asymptotic formula for the first moment of the quadratic Dirichlet L-functions over the rational function field, running over all monic, square-free polynomials of even degree at the central point. With some extra technical difficulties that doesn't appear in Florea's paper, we prove that there are extra main terms of size $(2g+2)q^{\frac{2g+2}{3}}, q^{\frac{g}{6}+\left[\frac{g}{2}\right]}$ and $q^{\frac{g}{6}+\left[\frac{g-1}{2}\right]}$, whilst bounding the error term by $q^{\frac{g}{2}(1+ε)}$.

math.NT↗