Roots of L-functions of characters over function
fields, generic linear independence and biases

Perret-Gentil, Corentin

doi:10.2140/ant.2020.14.1291

Cited by 6 publications

(6 citation statements)

References 51 publications

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“…Let us first state the large sieve bound in Theorem 2.21 for a general setting, using Perret–Gentil's improvement of Kowalski's large sieve for Frobenius [36, Theorem 5.14.(ii). (c)] and later apply it to our setting in Proposition 2.22.…”

Section: Preliminary Results and Notationsmentioning

confidence: 99%

“…To prove this theorem, we follow closely Kowalski's method based on the large sieve for Frobenius developed in [32] (and improved by Perret–Gentil [36]). The theorem above should be compared to Kowalski's bound (), which we now state.…”

Section: Introductionmentioning

confidence: 99%

“…Such a phenomenon was already present in [32, Corollary 8.10] where it results in a larger exponent of

q

in the multidimensional case. In our case, we keep the small

q

exponent, but the uniformity in

p

is lost when applying the improved bound [36, Theorem 5.14.(ii). (c)].…”

Section: Introductionmentioning

confidence: 99%

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Exceptional biases in counting primes over function fields

Bailleul,

Devin,

Keliher

et al. 2024

Journal of London Math Soc

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We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

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Section: Preliminary Results and Notationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exceptional biases in counting primes over function fields

Bailleul,

Devin,

Keliher

et al. 2024

Journal of London Math Soc

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“…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%

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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“…Several related results about the distribution of Kloosterman and Birch sums have recently been given by Perret-Gentil [19].…”

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confidence: 98%