Marc Technau
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View article: Zero sums amongst roots and Cilleruelo’s conjecture on the LCM of polynomial sequences
Zero sums amongst roots and Cilleruelo’s conjecture on the LCM of polynomial sequences Open
We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial f on the additional hypothesis that the polynomial be even. This strengthens earlier work of Rudni…
View article: Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences
Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences Open
We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial $f$ on the additional hypothesis that the polynomial be even. This strengthens earlier work of Rud…
View article: On Restricted Averages of Dedekind Sums
On Restricted Averages of Dedekind Sums Open
We investigate the averages of Dedekind sums over rational numbers in the set $\mathscr {F}_{\alpha }(Q) = \{\,{v}/{w}\in \mathbb {Q}: 0<w\leq Q\,\}\cap \lbrack 0, \alpha )$ for fixed $\alpha \leq 1/2$. In previous work, we obtained asy…
View article: Galois groups of $\binom{n}{0} + \binom{n}{1} X + \ldots + \binom{n}{6} X^6$
Galois groups of $\binom{n}{0} + \binom{n}{1} X + \ldots + \binom{n}{6} X^6$ Open
We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many $n$. This complements earlier work of Filaseta and Moy, who studied Galois groups of $\binom{n}…
View article: Remark on the Farey fraction spin chain
Remark on the Farey fraction spin chain Open
In 1999, Kleban and Özlük introduced a `Farey fraction spin chain' and made a conjecture regarding its asymptotic number of states with given energy, the latter being given (up to some normalisation) by the number $Φ(N)$ of $2\times2$ matr…
View article: On restricted averages of Dedekind sums
On restricted averages of Dedekind sums Open
We investigate the averages of Dedekind sums over rational numbers in the set $$\mathscr{F}_α(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0
View article: On polynomials with roots modulo almost all primes
On polynomials with roots modulo almost all primes Open
Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic qua…
View article: Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums
Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums Open
We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval $(0,1/2)$, establishing that they behave diffe…
View article: On polynomials with roots modulo almost all primes
On polynomials with roots modulo almost all primes Open
Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic qua…
View article: On the Distribution of <i>αp</i> Modulo One in Quadratic Number Fields
On the Distribution of <i>αp</i> Modulo One in Quadratic Number Fields Open
We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and…
View article: On the distribution of $\alpha p$ modulo one in quadratic number fields
On the distribution of $\alpha p$ modulo one in quadratic number fields Open
We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponen…
View article: Metric results on summatory arithmetic functions on Beatty sets
Metric results on summatory arithmetic functions on Beatty sets Open
Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(α) = \lbrace\, \lfloor nα\rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $α$, where $\lfloorξ\rfloor$ denotes t…
View article: Kloosterman sums with twice-differentiable functions
Kloosterman sums with twice-differentiable functions Open
We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2πi (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition…
View article: Generalised Beatty sets
Generalised Beatty sets Open
Generalised Beatty sets, that is, sets of the form { mα 1 + nα 2 + β : m, n ∈ N}, are studied, where ξ denotes the largest integer less than or equal to ξ.Such sets are shown to be contained in a suitable ordinary Beatty set { nα + β : n ∈…
View article: On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one
On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one Open
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 the ring of integers $\mathcal{O} = \mathbb{Z}[…
View article: On the distribution of $αp$ modulo one in imaginary quadratic number fields with class number one
On the distribution of $αp$ modulo one in imaginary quadratic number fields with class number one Open
We investigate the distribution of $αp$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[ω]$ of…
View article: On Beatty sets and some generalisations thereof
On Beatty sets and some generalisations thereof Open
Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves t…
View article: A Loewner Equation for Infinitely Many Slits
A Loewner Equation for Infinitely Many Slits Open
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 certain normalisation. We prove that a multiple slit Loewner…