arXiv subjects

Mounir Hayani

Publications and source records attributed to Mounir Hayani.

5 recordsLinked to original sources

arXiv2026

On the role of higher roots in prime ideal races

Let $L/K$ be a Galois extension of number fields, and let $C_1, C_2$ be conjugacy classes of $\mathrm{Gal}(L/K)$. By introducing algebraic parameters related to $C_1$ and $C_2$, we provide a conditional characterization for Chebyshev's bias logarithmic density $δ_{L/K}(C_1,C_2)$ to lie in the interval $(1/2, 1)$, meaning that the prime ideal race is biased. Unlike existing examples in the literature, where the bias comes from a difference in the number of square roots, the order of vanishing of Artin $L$-functions at $s=1/2$, or a combination of both, we use this criterion to construct Galois extensions where neither of these aspects plays a role. In our constructions, the bias arises entirely from a difference in the number of $2p$-th roots for an odd prime $p$. We prove that the minimal Galois group order required for this phenomenon is $96$ for $p=3$ and $320$ for $p=5$, where in both cases the group structure is a direct product of two generalized quaternion groups. Furthermore, we provide generalized constructions for all odd primes. Finally, these same algebraic parameters enable us to prove that an estimate established by Aoki and Koyama under the Deep Riemann Hypothesis holds unconditionally in certain cases.

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arXiv2026

Chebyshev's bias without linear independence

We confirm Chebyshev's observation that primes are strikingly more abundant in non-square residue classes modulo a fixed integer under the Generalized Riemann Hypothesis (GRH) by proving a (natural) density $1$ statement for prime counting functions in residue classes where each prime is weighted by its inverse square root. In contrast to the majority of the existing literature on the subject, we do not need to restrict to logarithmic densities to measure Chebyshev's bias, and we do not rely on any hypothesis on the zeros of $L$-functions that is stronger than GRH. Note: The same type of results presented here were independently proved by Arshay Sheth (2025) in the general context of automorphic forms, which implies our main asymptotic. While the spirit of the proofs is similar, Sheth develops an explicit formula for the partial Euler product and uses a result due to Gallagher (1980) to prove that some estimates hold outside a set of finite logarithmic measure. We share this independent work because it provides a completely self-contained and elementary proof relying only on the usual explicit formula, and it yields explicit error terms rather than an implicit $o(1)$ asymptotic.

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arXiv2026

A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races

We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races.

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arXiv2024

Disproving a weaker form of Hooley's conjecture

Hooley conjectured that $G(x;q) \ll x\log q$, as soon as $q\to +\infty$, where $G(x;q)$ represents the variance of primes $p \leq x$ in arithmetic progressions modulo $q$, weighted by $\log p$. In this paper, we study $G_η(x;q)$, a function similar to $G(x;q)$, but including the weighting factor $η\left(\frac{p}{x}\right)$, which has a dampening effect on the values of $G_η$. Our study is motivated by the disproof of Hooley's conjecture by Fiorilli and Martin in the range $q \asymp \log \log x$. Even though this weighting factor dampens the values, we still prove that an estimation of the form $G_η(x;q) \ll x\log q$ is false in the same range.

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arXiv2024

On the influence of the Galois group structure on the Chebyshev bias in number fields

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois theoretic statement showing a dichotomy between ``extreme Chebyshev biases'' and ``equal prime ideal counting''. We further introduce a group theoretic property that implies extreme biases. In the case of abelian extensions this leads to a complete characterization of Galois groups enabling extreme biases. In the case where the Galois group is a $p$-group, a simple criterion is deduced for the existence of extreme biases, and associated effective statements of Linnik type are obtained.

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