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pro vyhledávání: '"TRUDGIAN, TIM"'
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:
Jarso, Tamiru, Trudgian, Tim
For $q$ an odd prime power, we prove that there are always four consecutive primitive elements in the finite field $\mathbb{F}_{q}$ when $q> 2401$.
Externí odkaz:
http://arxiv.org/abs/2109.11691
Autor:
Platt, Dave, Trudgian, Tim
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$. That is, all zeroes $\beta + i\gamma$ of the Riemann zeta-function with $0<\gamma\leq 3\cdot 10^{12}$ have $\beta =
Externí odkaz:
http://arxiv.org/abs/2004.09765
Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that $R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that $R_k(x)=O(x^{\frac{1}{2k}+\epsilon})$. By esta
Externí odkaz:
http://arxiv.org/abs/1912.04972
Let $V(T)$ denote the number of sign changes in $\psi(x) - x$ for $x\in[1, T]$. We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, where $\gamma_{1} = 14.13\ldots$ denotes the ordinate of the lowest-l
Externí odkaz:
http://arxiv.org/abs/1910.14203
We provide an explicit estimate on the least primitive root mod $p^{2}$. We show, in particular, that every prime $p$ has a primitive root mod $p^{2}$ that is less than $p^{0.99}$.
Comment: 8 pages
Comment: 8 pages
Externí odkaz:
http://arxiv.org/abs/1908.11497
Autor:
McGown, Kevin J., Trudgian, Tim
We give a method for producing explicit bounds on $g(p)$, the least primitive root modulo $p$. Using our method we show that $g(p)<2r\,2^{r\omega(p-1)}\,p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$ where $r\geq 2$ is an integer parameter. This resul
Externí odkaz:
http://arxiv.org/abs/1904.12373
Autor:
Kobayashi, Mits, Trudgian, Tim
We show that the natural density of positive integers $n$ for which $\sigma(2n+1)\geq \sigma(2n)$ is between $0.053$ and $0.055$.
Externí odkaz:
http://arxiv.org/abs/1904.10064
Autor:
Morrill, Thomas, Trudgian, Tim
We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero $\beta$ with $1-
Externí odkaz:
http://arxiv.org/abs/1811.12521