arXiv subjects

Forrest J. Francis

Publications and source records attributed to Forrest J. Francis.

4 recordsLinked to original sources

arXiv2022

Explicit Subconvexity Estimates for Dirichlet $L$-functions

Given a Dirichlet character $χ$ modulo $q$ and its associated $L$-function, $L(s,χ)$, we provide an explicit version of Burgess' estimate for $|L(s, χ)|$. We use partial summation to provide bounds along the vertical lines $\Re{s} = 1 - {r}^{-1}$, where $r$ is a parameter associated with Burgess' character sum estimate. These bounds are then connected across the critical strip using the Phragmén--Lindelöf principle. In particular, for $σ\in [\frac{1}{2}, \frac{9}{10}]$, we establish $$|L(σ+ it, χ)| \leq (1.105) (0.692)^σq^{\frac{31}{80}-\frac{2}{5}σ}(\log{q})^{\frac{33}{16}-\frac{9}{8}σ} |σ+ it|.$$

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arXiv2020

Additive Representations of Natural Numbers

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

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arXiv2019

An Investigation Into Several Explicit Versions of Burgess' Bound

Let $χ$ be a Dirichlet character modulo $p$, a prime. In applications, one often needs estimates for short sums involving $χ$. One such estimate is the family of bounds known as \emph{Burgess' bound}. In this paper, we explore several minor adjustments one can make to the work of Enrique Treviño on explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a $k$th power non-residue modulo $p$ which is less than $p^α$ for several fixed $α$. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields.

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arXiv2018

Euler's Function on Products of Primes in Progressions

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $φ(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(e^{2πi/q})$ is true if and only if for all integers $k\geq 1$ we have \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,1)}.\] Here $\bar{N}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equiv 1~({\rm mod}~{q})$ and $C(q, 1)$ is the constant appearing in the asymptotic formula \[\prod_{\substack{p \leq x \\ p \equiv 1~({\rm mod}~{q})}} \left(1 - \frac{1}{p}\right) \sim \frac{C(q, 1)}{(\log{x})^\frac{1}{φ(q)}},\] as $x\rightarrow\infty$. We also prove that, for $q\leq 400,000$ and integers $a$ coprime to $q$, the analogous inequality \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,a)}\] holds for infinitely many values of $k$. If in addition $a$ is a not a square modulo $q$, then there are infinitely many $k$ for which this inequality holds and also infinitely many $k$ for which this inequality fails.

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