arXiv subjects

Nicol Leong

Publications and source records attributed to Nicol Leong.

9 recordsLinked to original sources

arXiv2026

Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta function

In this article, we give explicit bounds of order $\log t$ for $σ$ close to $1$, for two quantities: $|ζ'(σ+it)/ζ(σ+it)|$ and $|1/ζ(σ+it)|$. We correct an error in the literature, and especially in the case of $|1/ζ(σ+it)|$, also provide improvements in the constants. Using an argument involving the trigonometric polynomial, we additionally provide a slight asymptotic improvement within the classical zero-free region: $1/ζ(σ+it) \ll (\log t)^{11/12}$. The same method applied to the Korobov--Vinogradov zero-free region gives a new record: the unconditional bound $1/ζ(σ+it) \ll (\log t)^{2/3}(\log\log t)^{1/4}$.

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arXiv2024

A note on trigonometric polynomials for lower bounds of $ζ(s)$

Non-negative trigonometric polynomials satisfying certain properties are employed when studying a number of aspects of the Riemann zeta function. When establishing zero-free regions in the critical strip, the classical polynomial $3+4\cos(θ)+\cos(2θ)$ used by de la Vallée Poussin has since been replaced by more beneficial polynomials with larger degree. The classical polynomial was also employed by Titchmarsh to provide a lower bound on $|ζ(σ+it)|$ when $σ>1$. We show that this polynomial is optimal for this purpose.

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arXiv2024

New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{Ω(n)}, \end{equation*} where $Ω(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular, that $|W(x)|<2260x$ for all $x\geq 1$. This result provides new progress towards a conjecture of Sun, which asks whether $|W(x)| 0$.

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arXiv2024

Explicit bounds for the Riemann zeta-function on the 1-line

Explicit estimates for the Riemann zeta-function on the $1$-line are derived using various methods, in particular van der Corput lemmas of high order and a theorem of Borel and Carathéodory.

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arXiv2024

Explicit estimates for the Riemann zeta function close to the $1$-line

We provide explicit upper bounds of the order $\log t/\log\log t$ for $|ζ'(s)/ζ(s)|$ and $|1/ζ(s)|$ when $σ$ is close to $1$. These improve existing bounds for $ζ(s)$ on the $1$-line.

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arXiv2022

New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function

IWe prove new unconditional and explicit upper bounds for the reciprocal of the Riemann zeta function in the critical strip, of the orders $(\log t)^{11/12}$, $(\log t)^{11/12}(\log\log t)^{-3/4}$, and $(\log t)^{2/3}(\log\log t)^{1/4}$; these hold in prescribed classical, Littlewood, and Korobov--Vinogradov zero-free regions, respectively. From these bounds, we also derive new unconditional and explicit upper bounds for the Mertens function $M(x)$, of the orders $x (\log x)\exp\!\left(-\eta_1 \sqrt{\log x}\right)$, $x (\log x) \exp\!\left(-\eta_2 \sqrt{\log x \log\log x}\right)$, and $x (\log x)\exp\!\left(-\eta_3 (\log x)^{3/5}(\log\log x)^{-1/5}\right)$, for suitable constants $\eta_1,\eta_2,\eta_3>0$. A key feature of our approach is the use of computational smoothing, which ensures the resulting numerical bounds are strong.

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arXiv2022

Primitive elements with prescribed traces

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero element $ξ\in \mathbb{F}_{q^{n}}$ such that $ξ+ ξ^{-1}$ is primitive and $T(ξ)=a, T(ξ^{-1})=b$, where $T(ξ)$ denotes the trace of $ξ$ in $\mathbb{F}_{q}$. This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of $n \geq 5$. A related study involves the cubic extension $\mathbb{F}_{q^{3}}$ of $\mathbb{F}_{q}$. We show that if $q\geq 8\cdot 10^{12}$ then, for any $a\in \mathbb{F}_{q}$ we can find a primitive element $ξ\in \mathbb{F}_{q^{3}}$ such that $ξ+ ξ^{-1}$ is also a primitive element of $\mathbb{F}_{q^{3}}$, and for which the trace of $ξ$ is equal to $a$. The improves a result of Cohen and Gupta. Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions.

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arXiv2022

Primitive element pairs with a prescribed trace in the cubic extension of a finite field

We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.

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arXiv2017

Sums of reciprocals and the three distance theorem

In this paper we investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of $\{nα-γ\}^{-1}$, where $α$ and $γ$ are real parameters and $\{\,\cdot\,\}$ is the fractional part of a real number. Bounds for these sums have been studied for a long while in connection with various applications. In this paper we develop an alternative technique for obtaining upper bounds for the sums and obtain new efficient and fully explicit results. The technique uses the so-called three distance theorem.

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