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Publications (14)

math.NT2024

Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences

Marc Technau

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 t…

math.NT2019

The maximal order of iterated multiplicative functions

Christian Elsholtz, Marc Technau, Niclas Technau

Following Wigert, various authors, including Ramanujan, Gronwall, ErdÅs, IviÄ, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, gener…

math.CV2026

Shapiro's problem on polynomials with large partial sums of coefficients

Marc Technau

Given a polynomial $\sum_{I^} ½ a_{I^} ½ X^{I^} ½$ of degree $< d$ , bounded by one on the unit disk, how large can $∣ a_{0} + a_{1} + \dots + a_{n} ∣$ ( $n < d$ ) get? This question dates back at leas…

math.NT2023

Galois groups of $(0 n) + (1 n) X + \dots + (6 n) X^{6}$

Benjamin Klahn, Marc Technau

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 o…

math.NT2021

On the distribution of $I^\pm p$ modulo one in imaginary quadratic number fields with class number one

Stephan Baier, Marc Technau

We investigate the distribution of $I^\pm p$ modulo one in imaginary quadratic number fields $K \subset C$ with class number one, where $p$ is restricted to prime eleme…

math.NT2019

Kloosterman sums with twice-differentiable functions

Igor E. Shparlinski, Marc Technau

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2Ïi (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differe…

math.NT2022

On polynomials with roots modulo almost all primes

Christian Elsholtz, Benjamin Klahn, Marc Technau

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…

math.NT2023

On restricted averages of Dedekind sums

Paolo Minelli, Athanasios Sourmelidis, Marc Technau

We investigate the averages of Dedekind sums over rational numbers in the set $F_{I^} \pm (Q) := {v / w \in Q : 0 < w \leq Q} \cap [0, I^\pm)$ for fixed $I^\pm \leq 1/2$ . I…

math.NT2022

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Paolo Minelli, Athanasios Sourmelidis, Marc Technau

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…

math.NT2020

Metric results on summatory arithmetic functions on Beatty sets

Marc Technau, Agamemnon Zafeiropoulos

Let $f : N \to C$ be an arithmetic function and consider the Beatty set $B (I^\pm) = {⌊ n I^\pm ⌋ : n \in N}$ assoc…

math.NT2023

Remark on the Farey fraction spin chain

Marc Technau

In 1999, Kleban and ÃzlÃ¼k introduced a `Farey fraction spin chain' and made a conjecture regarding its asymptotic number of states with given energy, the latter being given (up t…

math.NT2021

On the distribution of $I^\pm p$ modulo one in quadratic number fields

Stephan Baier, Dwaipayan Mazumder, Marc Technau

We investigate the distribution of $I^\pm p$ modulo one in quadratic number fields $K$ with class number one, where $p$ is restricted to prime elements in the ring of integers…

math.NT2019

Modular hyperbolas and Beatty sequences

Marc Technau

Bounds for $max {m, m ~}$ subject to $m, m ~ \in Z \cap [1, p)$ , $p$ prime, $z$ indivisible by $p$ , $m m ~ \equiv z mod p$ and $m$ belonging to some fixed Be…

math.NT2016

The Least Prime Number in a Beatty Sequence

JÃ¶rn Steuding, Marc Technau

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.

Publications (14)

Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences

The maximal order of iterated multiplicative functions

Shapiro's problem on polynomials with large partial sums of coefficients

Galois groups of (0n​)+(1n​)X+…+(6n​)X6

On the distribution of I^±p modulo one in imaginary quadratic number fields with class number one

Kloosterman sums with twice-differentiable functions

On polynomials with roots modulo almost all primes

On restricted averages of Dedekind sums

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

Metric results on summatory arithmetic functions on Beatty sets

Remark on the Farey fraction spin chain

On the distribution of I^±p modulo one in quadratic number fields

Modular hyperbolas and Beatty sequences

The Least Prime Number in a Beatty Sequence

Galois groups of $(0 n) + (1 n) X + \dots + (6 n) X^{6}$

On the distribution of $I^\pm p$ modulo one in imaginary quadratic number fields with class number one

On the distribution of $I^\pm p$ modulo one in quadratic number fields