Zobrazeno 1 - 10
of 129
pro vyhledávání: '"11Y70"'
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:
Luo, Qi, Ye, Yangbo
Let $\lambda(n)$ and $\mu(n)$ denote the Liouville function and the M\"obius function, respectively. In this study, relationships between the values of $\lambda(n)$ and $\lambda(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are explored. Chowla's co
Externí odkaz:
http://arxiv.org/abs/2401.18082
Autor:
Shallue, Andrew, Webster, Jonathan
We report that there are $49679870$ Carmichael numbers less than $10^{22}$ which is an order of magnitude improvement on Richard Pinch's prior work. We find Carmichael numbers of the form $n = Pqr$ using an algorithm bifurcated by the size of $P$ wit
Externí odkaz:
http://arxiv.org/abs/2401.14495
Autor:
Schlitt, Jeremy
We examine a bias towards the zero residue class for the integers represented by binary quadratic forms. In many cases, we are able to prove that the bias comes from a secondary term in the associated asymptotic expansion (unlike Chebyshev's bias, wh
Externí odkaz:
http://arxiv.org/abs/2308.13959
Autor:
Holtz, Olga
Publikováno v:
Notices Amer. Math. Soc. 71 (2024), no. 6, 725-731
This short note for non-experts means to demystify the tasks of evaluating the Riemann Zeta Function at non-positive integers and at even natural numbers, both initially performed by Leonhard Euler. Treading in the footsteps of G. H. Hardy and others
Externí odkaz:
http://arxiv.org/abs/2308.11637
Autor:
Mok, Chung Pang, Zheng, Huimin
In this paper we propose conjectures that assert that, the sequence of Frobenius angles of a given elliptic curve over $\mathbf{Q}$ without complex multiplication is pseudorandom, in other words that the Frobenius angles are statistically independent
Externí odkaz:
http://arxiv.org/abs/2301.12823
Autor:
Johnston, Daniel R., Yang, Andrew
By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all $x\geq 2$ that
Externí odkaz:
http://arxiv.org/abs/2204.01980
Autor:
de Reyna, Juan Arias
(This is only a first preliminary version, any suggestions about it will be welcome.) In this paper it is shown how to compute Riemann's zeta function $\zeta(s)$ (and Riemann-Siegel $Z(t)$) at any point $s\in\mathbf C$ with a prescribed error $\varep
Externí odkaz:
http://arxiv.org/abs/2201.00342
Let $f(n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum_{n\leq x} f(n)$. Then $F(x)/\sqrt{x} = a\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in the limit,
Externí odkaz:
http://arxiv.org/abs/2112.05227
Autor:
Orlov, Alexey
We will generalize the combinatorial algorithms for computing $\pi(x)$ to compute sums ${F(x) = \sum_{p \leq x} p^k}$ for $k \in \mathbb{Z}_{\geq 0}$. The detailed exposition of algorithms is included along with implementation details.
Externí odkaz:
http://arxiv.org/abs/2111.15545