Zobrazeno 1 - 10
of 37
pro vyhledávání: '"Järviniemi, Olli"'
Autor:
Järviniemi, Olli, Hubinger, Evan
We study the tendency of AI systems to deceive by constructing a realistic simulation setting of a company AI assistant. The simulated company employees provide tasks for the assistant to complete, these tasks spanning writing assistance, information
Externí odkaz:
http://arxiv.org/abs/2405.01576
Autor:
Järviniemi, Olli, Teräväinen, Joni
Publikováno v:
Rev. Mat. Iberoam. 40(4), 1293-1350, 2024
We show that almost all sectors of the disc $\{z \in \mathbb{C}: |z|^2\leq X\}$ of area $(\log X)^{15.1}$ contain products of exactly two Gaussian primes, and that almost all sectors of area $(\log X)^{1 + \varepsilon}$ contain products of exactly th
Externí odkaz:
http://arxiv.org/abs/2303.05822
Autor:
Järviniemi, Olli
We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+\epsilon}$$ and $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon},$$ where $p_n$ is the $n
Externí odkaz:
http://arxiv.org/abs/2212.10965
This work concerns Artin's Conjecture on primitive roots and related problems for number fields. Let $K$ be a number field and let $W_1$ to $W_n$ be finitely generated subgroups of $K^\times$ of positive rank. We consider the index map, which maps a
Externí odkaz:
http://arxiv.org/abs/2211.15614
Autor:
Järviniemi, Olli, Perucca, Antonella
Since Hooley's seminal 1967 resolution of Artin's primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously conside
Externí odkaz:
http://arxiv.org/abs/2202.11329
Autor:
Järviniemi, Olli
For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always almost max
Externí odkaz:
http://arxiv.org/abs/2106.09813
Autor:
Järviniemi, Olli
Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree $d$ whose Galois group is $S_d$. Let $(a_n)$ be a linearly recuresive sequence of integers which has $P$ as its characteristic polynomial. We prove, under the general
Externí odkaz:
http://arxiv.org/abs/2102.04042
Autor:
Järviniemi, Olli, Teräväinen, Joni
Publikováno v:
Adv. Math., 429, 109187, 2023
A well-known open problem asks to show that $2^n+5$ is composite for almost all values of $n$. This was proposed by Gil Kalai as a possible Polymath project, and was posed originally by Christopher Hooley. We show that, assuming GRH and a form of the
Externí odkaz:
http://arxiv.org/abs/2010.01789
Autor:
Järviniemi, Olli
The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime divisors of some
Externí odkaz:
http://arxiv.org/abs/2006.00941
Autor:
Järviniemi, Olli
We determine a necessary and sufficient condition for the infinitude of primes $p$ such that none of the equations $a_i^x \equiv b_i \pmod{p}, 1 \le i \le n,$ are solvable. We control the insolvability of $a^x \equiv b \pmod{p}$ by power residues for
Externí odkaz:
http://arxiv.org/abs/1912.02526