arXiv subjects

Laurent Habsieger

Publications and source records attributed to Laurent Habsieger.

5 recordsLinked to original sources

arXiv2020

Explicit Asymptotics for Signed Binomial Sums and Applications to Carnevale-Voll Conjecture

Carnevale and Voll conjectured that j (--1) j $λ$ 1 j $λ$ 2 j = 0 when $λ$ 1 and $λ$ 2 are two distinct integers. We check the conjecture when either $λ$ 2 or $λ$ 1 -- $λ$ 2 is small. We investigate the asymptotic behaviour of their sum when the ratio r := $λ$ 1 /$λ$ 2 is fixed and $λ$ 2 goes to infinity. We find an explicit range r $\ge$ 5.8362 on which the conjecture is true. We show that the conjecture is almost surely true for any fixed r. For r close to 1, we give several explicit intervals on which the conjecture is also true.

math.CO↗

arXiv2019

Explicit Bounds For The Diophantine Equation A!B! = C!

A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A $\le$ B $\le$ C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, such as the one given by Erd{ö}s C -- B $\le$ 5 log log C, or the one given by Bhat and Ramachandra C --B $\le$ (1/ log 2+o(1)) log log C. We check the conjecture for B $\le$ 10 3000 and give better explicit bounds such as C -- B $\le$ log log(B+1) log 2 -- 0.8803.

math.NT↗

arXiv2016

Sums of the digits in bases 2 and 3

Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N 0.970359. The proof only uses the separate (or marginal) distributions of the values of s 2 (n) and s 3 (n).

math.NT↗

arXiv2016

A numerical note on upper bounds for b 2 [g] sets

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates on the size of a B 2 [g] set in an interval of integers in the cases g = 2, 3, 4 and 5.

math.NT↗

arXiv2011

Spiegelungssatz: a combinatorial proof for the 4-rank

The Spiegelungssatz is an inequality between the (4)-ranks of the narrow ideal class groups of the quadratic fields (\mathbb{Q}(\sqrt{D})) and (\mathbb{Q}(\sqrt{-D})). We provide a combinatorial proof of this inequality. Our interpretation gives an affine system of equations that allows to describe precisely some equality cases.

math.NT↗