Chebyshev’s bias in dihedral and generalized
quaternion Galois groups

Bailleul, Alexandre

doi:10.2140/ant.2021.15.999

Cited by 5 publications

(14 citation statements)

References 26 publications

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“…Combining this result with (2.3) and using that 𝜙(𝑚) = |𝑚| 1+𝑜 (1) we deduce (1.2). We now state our result for 𝛿 𝑚;𝑎 1 ,…,𝑎 𝑟 when 𝑟 ⩾ 3.…”

Section: Lemma 22 the Entries Of Covmentioning

confidence: 55%

“…The assumptions they used are the generalized Riemann hypothesis (GRH) and the linear independence hypothesis (LI) (which is the assumption that the non-negative imaginary parts of the zeros of all Dirichlet 𝐿-functions attached to all Dirichlet characters modulo 𝑞 are linearly independent over ℚ). They proved, under these two hypotheses, that 𝛿(𝑞; 𝑎, 𝑏) = 1 2 if 𝑎 and 𝑏 are both quadratic residues or quadratic non-residues modulo 𝑞, and otherwise that 𝛿(𝑞; 𝑎, 𝑏) > 1 2 if 𝑎 is a quadratic non-residue and 𝑏 is a quadratic residue modulo 𝑞, thus confirming Chebyshev's observation. We mention the interesting articles of Granville and Martin [11] and Ford and Konyagin [8] for detailed review about the history of this subject.…”

Section: Introductionmentioning

confidence: 65%

“…Cha, Fiorilli, and Jouve studied the prime number race for elliptic curves over the function field of a proper, smooth, and geometrically connected curve over a finite field. Further related subjects have been studied since then in [7], [6], [4], [16], and [1].…”

Section: Introductionmentioning

confidence: 99%

“…Feuerverger and Martin [9] were the first to compute densities with three competitors. They showed, under GRH and LI, that 𝛿(8; 3, 5, 7) = 𝛿(8; 7, 5, 3) ≠ 1 3! even if 3, 5, and 7 are quadratic non-residues modulo 8.…”

Section: Introductionmentioning

confidence: 99%

“…Under the assumption of LI, we know from [2, Theorem 6.1] that if 𝑎 1 and 𝑎 2 are both quadratic residues or both quadratic non-residues modulo 𝑚, then 𝛿 𝑚;𝑎 1 ,𝑎 2 = 𝛿 𝑚;𝑎 2 ,𝑎 1 = 1 2 . We say in this case that the race {𝑚; 𝑎 1 , 𝑎 2 } is unbiased.…”

Section: Introductionmentioning

confidence: 99%

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Inequities in the Shanks–Renyi prime number race over function fields

Sedrati

2022

Mathematika

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Fix a prime p>2$p >2$ and a finite field double-struckFq$\mathbb {F}_{q}$ with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring Fq[T]$\mathbb {F}_{q}[T]$ such that prefixdegfalse(mfalse)$\deg (m)$ is large. Fix an integer r⩾2$r\geqslant 2$, and let a1,⋯,ar$a_1,\dots ,a_r$ be distinct residue classes modulo m that are relatively prime to m. In this paper, we derive an asymptotic formula for the natural density δm;a1,⋯,ar$\delta _{m;a_1,\dots ,a_r}$ of the set of all positive integers X such that ∑N=1Xπq(a1,m,N)>∑N=1Xπq(a2,m,N)>⋯>∑N=1Xπq(ar,m,N)$\sum _{N=1}^{X} \pi _{q}(a_1,m,N) > \sum _{N=1}^{X} \pi _{q}(a_2,m,N)>\cdots > \sum _{N=1}^{X} \pi _{q}(a_r,m,N)$, where πq(ai,m,N)$\pi _{q}(a_i,m,N)$ denotes the number of irreducible monic polynomials in Fq[T]$\mathbb {F}_{q}[T]$ of degree N that are congruent to aimodm$ a_i \bmod m$, under the assumption of LI (Linear Independence Hypothesis). Many consequences follow from our results. First, we deduce the exact rate at which δm;a1,a2$\delta _{m;a_1,a_2}$ converges to 12$\frac{1}{2}$ as prefixdegfalse(mfalse)$\deg (m)$ grows, where a1 is a quadratic non‐residue and a2 is a quadratic residue modulo m, generalizing the work of Fiorilli and Martin. Furthermore, similarly to the number field setting, we show that two‐way races behave differently than races involving three or more competitors, once prefixdegfalse(mfalse)$\deg (m)$ is large. In particular, biases do appear in races involving three or more quadratic residues (or quadratic non‐residues) modulo m. This work is a function field analog of the work of Lamzouri, who established similar results in the number field case. However, we exhibit some examples of races in function fields where LI is false, and where the associated densities vanish, or behave differently than in the number field setting.

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“…Combining this result with (2.3) and using that 𝜙(𝑚) = |𝑚| 1+𝑜 (1) we deduce (1.2). We now state our result for 𝛿 𝑚;𝑎 1 ,…,𝑎 𝑟 when 𝑟 ⩾ 3.…”

Section: Lemma 22 the Entries Of Covmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inequities in the Shanks–Renyi prime number race over function fields

Sedrati

2022

Mathematika

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Chebyshev's bias against splitting and principal primes in global fields

Aoki

Koyama

2023

Journal of Number Theory

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Distribution of Frobenius Elements in Families of Galois Extensions

Fiorilli¹,

Jouve

2023

J. Inst. Math. Jussieu

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Given a Galois extension $L/K$ of number fields, we describe fine distribution properties of Frobenius elements via invariants from representations of finite Galois groups and ramification theory. We exhibit explicit families of extensions in which we evaluate these invariants and deduce a detailed understanding and a precise description of the possible asymmetries. We establish a general bound on the generic fluctuations of the error term in the Chebotarev density theorem, which under GRH is sharper than the Murty–Murty–Saradha and Bellaïche refinements of the Lagarias–Odlyzko and Serre bounds, and which we believe is best possible (assuming simplicity, it is of the quality of Montgomery’s conjecture on primes in arithmetic progressions). Under GRH and a hypothesis on the multiplicities of zeros up to a certain height, we show that in certain families, these fluctuations are dominated by a constant lower order term. As an application of our ideas, we refine and generalize results of K. Murty and of Bellaïche, and we answer a question of Ng. In particular, in the case where $L/{\mathbb {Q}}$ is Galois and supersolvable, we prove a strong form of a conjecture of K. Murty on the unramified prime ideal of least norm in a given Frobenius set. The tools we use include the Rubinstein–Sarnak machinery based on limiting distributions and a blend of algebraic, analytic, representation theoretic, probabilistic and combinatorial techniques.

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Chebyshev’s bias in dihedral and generalized quaternion Galois groups

Cited by 5 publications

References 26 publications

Inequities in the Shanks–Renyi prime number race over function fields

Inequities in the Shanks–Renyi prime number race over function fields

Chebyshev's bias against splitting and principal primes in global fields

Distribution of Frobenius Elements in Families of Galois Extensions

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