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Loïc Grenié

Publications and source records attributed to Loïc Grenié.

16 recordsLinked to original sources
arXiv2026

Bounds for the residue of Dedekind zeta functions under GRH and applications

Let $κ_{\mathbb{K}}$ be the residue at $1$ of the Dedekind zeta function of a number field $\mathbb{K}$. We prove explicit upper and lower bounds for $κ_{\mathbb{K}}$ under the Generalized Riemann Hypothesis (GRH). Previous explicit bounds of comparable generality are weaker and require a weak form of Artin's holomorphy conjecture: our bounds improve the constants while assuming only GRH for the relevant Dedekind zeta functions. The new lower bound enables all imaginary quadratic fields with class number $\leq 2000$ to be detected (assuming GRH). Furthermore, we demonstrate that the upper bound can be significantly reduced when considering extensions of any fixed, nontrivial field.

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arXiv2025

A lower bound for the number of Egyptian fractions

An Egyptian fraction is a sum of the form $1/n_1 + \cdots + 1/n_r$ where $n_1, \dots, n_k$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $E_N$ of rational numbers that can be represented by Egyptian fractions with denominators not exceeding $N$. More precisely, we show that for every integer $k \geq 4$ such that $\ln_k N \geq 3/2$ it holds $$ \frac{\ln(|E_N|)}{\ln 2} \geq \Big(2 - \frac{3}{\ln_k N}\Big)\frac{N}{\ln N}\prod_{j=3}^{k} \ln_j N , $$ where $\ln_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős who established a similar bound but under the more stringent condition $\ln_k N\geq k$ and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of $|E_N|$ for large positive integers $N$, and we give a table of $|E_N|$ for $N$ up to $154$.

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arXiv2025

Breaking the 4 barrier for the bound of a generating set of the class group

Let $K$ be a field of degree $n$ and discriminant with absolute value $Δ$. Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of $K$ and prove that the norm of the ideals in that set is $\leq (4-1/(2n))\log^2Δ$, except for a finite number of fields of degree $n\leq 4$. For those fields, the conclusion holds with the slightly larger limit $(4-1/(2n)+1/(2n^2))\log^2Δ$. When the cardinality of $\mathcal C\!\ell$ is odd the bounds improve to $(4-2/(3n))\log^2Δ$, again with finitely many exceptions in degree $n\leq 4$, and to $(4-2/(3n)+3/(8n^2))\log^2Δ$ without exceptions.

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arXiv2019

Counting Egyptian fractions

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.

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arXiv2017

Explicit bounds for generators of the class group

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group $\mathcal C\!\ell_{\mathbf K}$ of a number field ${\mathbf K}$ may be generated using prime ideals whose norm is bounded by $12\log^2Δ_{\mathbf K}$, and by $(4+o(1))\log^2Δ_{\mathbf K}$ asymptotically, where $Δ_{\mathbf K}$ is the absolute value of the discriminant of ${\mathbf K}$. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates $\mathcal C\!\ell_{\mathbf K}$ and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that $\mathcal C\!\ell_{\mathbf K}$ is generated by prime ideals whose norm is bounded by the minimum of $4.01\log^2Δ_{\mathbf K}$, $4\big(1+\big(2πe^γ)^{-n_{\mathbf K}}\big)^2\log^2Δ_{\mathbf K}$ and $4\big(\logΔ_{\mathbf K}+\log\logΔ_{\mathbf K}-(γ+\log 2π)n_{\mathbf K}+1+(n_{\mathbf K}+1)\frac{\log(7\logΔ_{\mathbf K})}{\logΔ_{\mathbf K}}\big)^2$. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size $\asymp (\logΔ_{\mathbf K}\log\logΔ_{\mathbf K})^2$. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be experimentally smaller than $\log^2Δ_{\mathbf K}$ except for 7 out of 31000 fields.

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arXiv2016

An improvement to an algorithm of Belabas, Diaz y Diaz and Friedman

In [BDyDF08] Belabas, Diaz y Diaz and Friedman show a way to determine, assuming the Generalized Riemann Hypothesis, a set of prime ideals that generate the class group of a number field. Their method is efficient because it produces a set of ideals that is smaller than earlier proved results. Here we show how to use their main result to algorithmically produce a bound that is lower than the one they prove.

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arXiv2015

Zeros of Dedekind zeta functions under GRH

Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.

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arXiv2015

Explicit smoothed prime ideals theorems under GRH

Let $ψ_{\mathbb K}$ be the Chebyshev function of a number field $\mathbb K$. Let $ψ^{(1)}_{\mathbb K}(x):=\int_{0}^{x}ψ_{\mathbb K}(t)\,d t$ and $ψ^{(2)}_{\mathbb K}(x):=2\int_{0}^{x}ψ^{(1)}_{\mathbb K}(t)\,d t$. We prove under GRH explicit inequalities for the differences $|ψ^{(1)}_{\mathbb K}(x) - \tfrac{x^2}{2}|$ and $|ψ^{(2)}_{\mathbb K}(x) - \tfrac{x^3}{3}|$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.

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