Chebyshev’s bias for analytic<i>L</i>-functions

Devin, Lucile

doi:10.1017/s0305004119000100

Cited by 20 publications

(37 citation statements)

References 37 publications

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“…elements of {γ ≥ 0 : L( 1 2 + iγ, χ) = 0} that are linearly independent of all the other elements of Z(q)). This condition has been further weakened in [Dev19] in the case of a prime number race between two contestants (D = 1). In this paper, we extend the weakening of LI for the weak inclusiveness of prime number races with any fixed number of contestants as a corollary of our main theorem (Corollary 2).…”

Section: Prime Number Racesmentioning

confidence: 99%

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Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

Devin¹

2020

Can. Math. Bull.

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We generalize current known distribution results on Shanks-Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let π(x; q, a) be the number of primes up to x that are congruent to a modulo q. For a fixed integer q and distinct invertible congruence classes a 0 , a 1 , . . . , a D , assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real x for which the inequalities π(x; q, a 0 ) > π(x; q, a 1 ) > . . . > π(x; q, a D ) are simultaneously satisfied admits a logarithmic density.

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Section: Prime Number Racesmentioning

confidence: 99%

“…The existence of the limiting logarithmic distribution of an almost-periodic function F with a finite number of terms, as in Theorem 1, is a consequence of the Kronecker-Weyl Equidistribution Theorem as in [Hum12,Lem. 4.3] (see also [Dev19,Lem. 4.8]).…”

Section: Prime Number Racesmentioning

confidence: 99%

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

Devin¹

2020

Can. Math. Bull.

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“…Let us now explain how this relates to biases and the random vectors X k,N puq defined above. We adapt classical arguments [RS94,MN17,Dev19] to the function field setting, as in [Cha08,DM18], to show: Theorem 1.14 (Limiting distribution, expected value). The random vector X k,N puq admits a compactly supported limiting distribution as N Ñ 8 with κ ă N {2 fixed.…”

Section: F P (Fixed)mentioning

confidence: 99%

“…Remark 1.18. Concerning the stronger properties of Theorem 1.15 (absolute continuity, symmetry), Devin [Dev19] and Martin-Ng [MN17] have shown that they hold under weaker conditions than full linear independence. However, we cannot exploit these here since their statements always involve all the roots/eigenvalues, while results obtained from the large sieve will be limited to a small subset.…”

Section: F P (Fixed)mentioning

confidence: 99%

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

Perret-Gentil

2019

Preprint

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We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field Fq, for almost all couples of arguments in F q , as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick-Waxman and Keating-Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenvalues of ℓ-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties.Remark 1.5. Applying Deligne's equidistribution theorem and [Kat88, Kat90] would show that ´fq n pa `b1 q, . . . , f q n pa `bt q ¯aPF q n , a`b i ‰0 converges in law (with respect to the uniform measure), as q n Ñ 8, to a random vector in Ω t r distributed with respect to the product measure ptr ˚µr q bt , when b i P F q n1 Here and from now on, δB will denote the Kronecker symbol with respect to a binary variable B, i.e. δB " 1 if B is true, 0 otherwise. In particular, r δ r odd is equal to r if the latter is odd, and to 1 otherwise.

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“…The literature on this question is rich, and much progress has been made in the recent years. We mention the works [Lit1, KT,Kac2,Pu,RbS,FM,La1] on the Shanks-Rényi problem, as well as generalizations over number fields [Maz,Sa1,Fi2,De,FoS,DGK,LOS,Me] and over function fields [Cha, CI, DM, CFJ]. For an exhaustive list of the numerous papers on the subject, see [GM, MS, M+].…”

mentioning

confidence: 99%

Distribution of Frobenius elements in families of Galois extensions

Fiorilli¹,

Jouve²

2020

Preprint

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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 J. Bellaïche and we answer a question of N. Ng. In particular, in the case where L/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 for analyticL-functions

Cited by 20 publications

References 37 publications

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

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

Distribution of Frobenius elements in families of Galois extensions

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