Patterns of primes in Chebotarev sets

Vatwani, Akshaa; Wong, Peng‐Jie

doi:10.1142/s1793042117500956

Cited by 7 publications

(17 citation statements)

References 10 publications

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“…Note that the original Green-Tao theorem automatically holds for all positive density subsets of the primes. Using work of Pintz [13] and its extension to Chebotarev sets by [15], our next result shows that we can find arbitrarily long arithmetic progressions within the primes in P C,α,β that are within a given distance from other primes also in P C,α,β . Corollary 1.4.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 87%

Primes with Beatty and Chebotarev conditions

Kazdan

McDonald

2020

Journal of Number Theory

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We study the prime numbers that lie in Beatty sequences of the form ⌊αn + β⌋ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence with α of finite type and a Chebotarev class of some Galois extension is precisely the product of the densities α −1 · |C| |G| . Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri-Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bounded gaps result and the Green-Tao theorem.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 87%

Primes with Beatty and Chebotarev conditions

Kazdan

McDonald

2020

Journal of Number Theory

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“…

M_{k} \geq \log k - c

for all

k \geq c

, for some absolute c . From this bound, Theorem , and a direct calculation (see [, Theorem 4.7]), we establish the following result. Theorem In the notation and assumptions as above, let

m \in N

and

H = {h_{1}, \dots, h_{k}}

be an admissible set with

k \geq ⌈ κ \exp false(2 m {(dim Π)}^{2} / θ false) ⌉

, for an absolute constant κ.…”

Section: Bounded Gaps Between Primes For Modular Formsmentioning

confidence: 88%

“…Now let us borrow a lower bound for M k from [37, Theorem 23], i.e. M k log k − c for all k c, for some absolute c. From this bound, Theorem 13, and a direct calculation (see [45,Theorem 4.7]), we establish the following result. THEOREM 14.…”

Section: Proofs Ofmentioning

confidence: 96%

“…We let

double-struckP

denote the set of primes, and we let the functions μ and

τ_{k} false(n false)

stand for the Möbius function and the number of representations of n as a product of k natural numbers, respectively. For the sake of convenience, we shall further introduce the following notation used in . For

n, N \in N

, we denote

N < n \leq 2 N

n \sim N

, and we define the radical of n by

rad false(n false) = \prod_{p | n} p

.…”

Section: Bounded Gaps Between Primes For Modular Formsmentioning

confidence: 99%

“…Now consider an abelian Galois extension K/F . As Artin reciprocity implies that each irreducible representation ρ of G = Gal(K/F ) corresponds to an idèle class character χ of F , from (45), it follows that…”

Section: An Automorphic Variant Of Dirichlet's Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Bombieri–vinogradov Theorems for Modular Forms and Applications

Wong

2019

Mathematika

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In this article, we consider a prime number theorem for arithmetic progressions “weighted” by Fourier coefficients of modular forms, and we develop Siegel‐Walfisz type and Bombieri–Vinogradov type estimates for such a modular analogue. As an application, we have a Turán type estimate for modular forms asserting that for any δ>0 and non‐CM normalised Hecke eigenform f, scriptPffalse(a,qfalse)≤q2+δ,with a possible exceptional set of q of density 0 (depending at most on f and δ), where (a,q)=1, scriptPffalse(a,qfalse) denotes the least prime p, with λffalse(pfalse)≠0, congruent to a(modq), and λffalse(pfalse) is the pth Fourier coefficient of f. Moreover, we show the existence of a positive absolute constant C0, independent of f, such that there are infinitely many pairs (p1,p2) of distinct primes satisfying false|p1−p2false|≤C0andλffalse(p1false)λffalse(p2false)≠0,which presents a modular analogue of the recent work of Maynard and Zhang on bounded gaps between primes.

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Bounded gaps between primes in short intervals

Alweiss

Luo

2018

Res. number theory

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Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form [x − x 0.525 , x] for large x. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any δ ∈ [0.525, 1] there exist positive integers k, d such that for sufficiently large x, the interval(log x) k pairs of consecutive primes differing by at most d. This confirms a speculation of Maynard that results on small gaps between primes can be refined to the setting of short intervals of this length.

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Patterns of primes in Chebotarev sets

Cited by 7 publications

References 10 publications

Primes with Beatty and Chebotarev conditions

Primes with Beatty and Chebotarev conditions

Bombieri–vinogradov Theorems for Modular Forms and Applications

Bounded gaps between primes in short intervals

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