Biases in the prime number race of function fields

Cha, Byungchul; Kim, Seick

doi:10.1016/j.jnt.2009.09.015

Cited by 6 publications

(3 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since then, other generalizations have been extensively studied by many authors. Cha and Seick [CK10] generalized the results of [FM00]. 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.…”

Section: Introductionmentioning

confidence: 90%

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

Sedrati¹

2021

Preprint

View full text Add to dashboard Cite

Fix a prime p > 2 and a finite field Fq with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring Fq[T ] such that deg(m) is large. Fix an integer r 2, and let a1, . . . , ar 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;a 1 ,...,ar of the set of all positive integers X such thatwhere πq(ai, m, N ) denotes the number of irreducible monic polynomials in Fq[T ] of degree N that are congruent to ai mod m, under the assumption of LI (Linear Independence Hypothesis). Many consequences follow from our results. First, we deduce the exact rate at which δm;a 1 ,a 2 converges to 1 2 as 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 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.

show abstract

Section: Introductionmentioning

confidence: 90%

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

Sedrati¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Proof of Theorem Since we know from [5, Theorem 2.3] that

\delta _{m;ba_1,\dots ,ba_r} = \delta _{m;a_1,\dots ,a_r}

for any residue class modulo m , then it is sufficient to construct quadratic residues

a_j

modulo m . In fact, we take

b_j = b a_j

for any b quadratic non‐residue modulo m to get the analogous result for quadratic non‐residues modulo m .…”

Section: Explicit Constructions Of Biased Racesmentioning

confidence: 99%

“…Since then, other generalizations have been extensively studied by many authors. Cha and Seick [5] generalized the results of [9]. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Sedrati

2022

Mathematika

View full text Add to dashboard Cite

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.

show abstract

Chebyshev's bias for products of irreducible polynomials

Devin

Meng

2021

Advances in Mathematics

View full text Add to dashboard Cite

Biases in the prime number race of function fields

Cited by 6 publications

References 5 publications

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

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 for products of irreducible polynomials

Contact Info

Product

Resources

About