Angles of Gaussian primes

Rudnick, Zeév; Waxman, Ezra

doi:10.1007/s11856-019-1867-5

Cited by 18 publications

(21 citation statements)

References 12 publications

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“…Similarly, in [PG19], Perret-Gentil proves several results on generic linear independence, related to the function field version of the bias in the distribution of angles of Gaussian primes in sectors (see also [RW19]) and to the distribution of irreducible polynomials in short intervals (see also [KR14]). However these results show the existence of small subsets of the zeros satisfying the linear independence, and this is not enough to apply Theorem 4 in the case D ≥ 2 : we would need to control the multiplicities on the whole set of zeros to conclude.…”

Section: Chebyshev's Bias In Function Fieldsmentioning

confidence: 90%

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: Chebyshev's Bias In Function Fieldsmentioning

confidence: 90%

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

Devin¹

2020

Can. Math. Bull.

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“…Therefore, for small intervals, one can expect at most a statistical statement about the existence of multiplier angles. We show in this paper that for a fixed K 1, nearly every interval of length 2π/K in (−π, π] contains a multiplier angle with the property that the absolute value of the multiplier is bounded above by a polynomial in K. Our work is inspired by recent results in analytic number theory, in particular by Rudnick-Waxman [RW19] and . Another equidistribution result on multipliers of hyperbolic rational maps was recently obtained by Sharp-Stylianou [SS].…”

mentioning

confidence: 83%

“…We will use the variance of Φ * K,t in Section 5 to prove Theorem 1.1. These definitions are inspired by the so-called smooth count of Gaussian primes introduced in [RW19].…”

Section: Dynamical L-functionsmentioning

confidence: 99%

Quantitative equidistribution of angles of multipliers

Nie

2022

Fund. Math.

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“…To prove this claim, we need an interesting elementary result on the repulsion of angles for a real quadratic field (Lemma 15), which shows that angles are well-spaced. This repulsion between angles is also an important fact when one studies angle distribution in short arcs (see [35] for the imaginary quadratic field Q(i)).…”

Section: Nodal Domainsmentioning

confidence: 99%

Sup-Norm and Nodal Domains of Dihedral Maass Forms

Huang

2019

Commun. Math. Phys.

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In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let φ be a dihedral Maass form with spectral parameter t φ , then we prove thatwhich is an improvement over the bound t 5/12+ε φ φ 2 given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindelöf Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than t 1/8−ε φ for any ε > 0 for almost all dihedral Maass forms. This is to be compared with the "local Weyl law", which gives λ 1/4 φ / log λ φ for any negative curvature surface (see [2]). Iwaniec-Sarnak [21] also proved the same bounds (1.1) for the Date: February 26, 2019.

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Angles of Gaussian primes

Cited by 18 publications

References 12 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

Quantitative equidistribution of angles of multipliers

Sup-Norm and Nodal Domains of Dihedral Maass Forms

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