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The least prime number in a Beatty sequence
Abstract We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.
Emptiness Problems for Integer Circuits
TLDR
It turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity. Expand
Metric results on summatory arithmetic functions on Beatty sets
Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to aExpand
Modular hyperbolas and Beatty sequences
Bounds for $\max\{m,\tilde{m}\}$ subject to $m,\tilde{m} \in \mathbb{Z}\cap[1,p)$, $p$ prime, $z$ indivisible by $p$, $m\tilde{m}\equiv z\bmod p$ and $m$ belonging to some fixed Beatty sequence $\{Expand
Emptiness problems for integer circuits
TLDR
It turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a well-studied, major open problem in algebraic computing complexity. Expand
A Loewner Equation for Infinitely Many Slits
It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certainExpand
Kloosterman sums with twice-differentiable functions.
We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differentiable functionExpand
The maximal order of iterated multiplicative functions
Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivic, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert'sExpand
On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one.
We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in theExpand
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