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pro vyhledávání: '"Leong, Nicol"'
In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}, \end{equation*} where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular, that $|W(
Externí odkaz:
http://arxiv.org/abs/2408.04143
Autor:
Leong, Nicol
In this article, we improve on two explicit bounds of order $\log t$ for $\sigma$ close to $1$: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and $|1/\zeta(\sigma +it)|$. Our method provides marked improvements in the constants, especially with regards to
Externí odkaz:
http://arxiv.org/abs/2405.04869
Autor:
Leong, Nicol, Mossinghoff, Michael J.
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(\theta)+\co
Externí odkaz:
http://arxiv.org/abs/2404.05928
Autor:
Cully-Hugill, Michaela, Leong, Nicol
We provide explicit upper bounds of the order $\log t/\log\log t$ for $|\zeta'(s)/\zeta(s)|$ and $|1/\zeta(s)|$ when $\sigma$ is close to $1$. These improve existing bounds for $\zeta(s)$ on the $1$-line.
Externí odkaz:
http://arxiv.org/abs/2312.09412
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 Carath\'{e}odory.
Comment: 31 pages
Comment: 31 pages
Externí odkaz:
http://arxiv.org/abs/2306.13289
Autor:
Lee, Ethan S., Leong, Nicol
In this paper, we establish new explicit bounds for the Mertens function $M(x)$. In particular, we compare $M(x)$ against a short-sum over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$, whose difference we can bound using recent compu
Externí odkaz:
http://arxiv.org/abs/2208.06141
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 $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{
Externí odkaz:
http://arxiv.org/abs/2202.00829
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 $\x
Externí odkaz:
http://arxiv.org/abs/2112.10268
Autor:
Cully-Hugill, Michaela, Leong, Nicol
Publikováno v:
In Journal of Mathematical Analysis and Applications 15 December 2024 540(2)
Autor:
Beresnevich, Victor, Leong, Nicol
In this paper we investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of $\{n\alpha-\gamma\}^{-1}$, where $\alpha$ and $\gamma$ are real parameters and $\{\,\cdot\,\}$ is the fractional part of a real number
Externí odkaz:
http://arxiv.org/abs/1712.03758