arXiv subjects

Korneel Debaene

Publications and source records attributed to Korneel Debaene.

4 recordsLinked to original sources

arXiv2017

Explicit counting of ideals and a Brun-Titchmarsh inequality for the Chebotarev Density Theorem

We prove a bound on the number of primes with a given splitting behaviour in a given field extension. This bound generalises the Brun-Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to $x$.

math.NT↗

arXiv2016

Dimensions of the irreducible representations of the symmetric and alternating group

We establish the existence of an irreducible representation of $A_n$ whose dimension does not occur as the dimension of an irreducible representation of $S_n$, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in the proof is a result on large prime factors in short intervals.

math.CO↗

arXiv2014

The first factor of the class number of the $p$-th cyclotomic field

Kummer's conjecture states that the relative class number of the $p$-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true -- it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer's conjecture, and more precisely we prove that $\log(h_p^-)=\frac{p+3}{4}\log(p) + \frac{p}{2}\log(2π)+\log(1-β)+O(\log_2 p)$, where $β$ is a possible Siegel zero of an $L(s,χ)$, $χ$ odd.

math.NT↗

arXiv2013

Asymptotic expansions, $L$-values and a new Quantum Modular Form

In 2010 Zagier introduced the notion of a quantum modular form. One of his first examples was the "strange" function $F(q)$ of Kontsevich. Here we produce a new example of a quantum modular form by making use of some of Ramanujan's mock theta functions. Using these functions and their transformation behaviour, we also compute asymptotic expansions similar to expansions of $F(q)$.

math.NT↗