On an explicit zero-free region for the Dedekind zeta-function

Lee, Ethan Simpson

doi:10.1016/j.jnt.2020.12.015

Cited by 8 publications

(3 citation statements)

References 9 publications

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“…There is no generalised Riemann height, so we can only use zero-free and zero-density regions of ζ K to obtain this estimate. At the time of writing, the second author provides the latest zero-free results in [23], and Trudgian provides the latest peer-reviewed zero-density results in [39], soon to be superseded by Hasanalizade et al [17].…”

Section: 2mentioning

confidence: 99%

Explicit Interval Estimates for Prime Numbers

Cully-Hugill¹,

Lee²

2021

Preprint

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Section: 2mentioning

confidence: 99%

Explicit Interval Estimates for Prime Numbers

Cully-Hugill¹,

Lee²

2021

Preprint

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“…The non-trivial zeros of ζ K (s) satisfy 0 < Re(s) < 1, and we note that there might exist a single, simple, real zero 0 < β 0 < 1, which is called an exceptional zero. Explicit bounds for β 0 may be found in [1,16,23].…”

Section: 1mentioning

confidence: 99%

Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias

Hathi¹,

Lee²

2021

Preprint

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The first result of our article is another proof of Mertens' third theorem in the generalised setting of number fields, which generalises a method of Hardy. Diamond and Pintz showed that the error term in the classic Mertens' third theorem changes sign infinitely often, so the second result of our article generalises Diamond and Pintz's result for number fields. That is, subject to some conditions, we show that the error term in Mertens' third theorem for number fields changes sign infinitely often. To prove this result, we also needed to prove Cramér's inequality for number fields, which is interesting in its own right. Lamzouri built upon Diamond and Pintz's work to prove the existence of the logarithmic density of the set of x such that the error term in Mertens' third theorem is positive, so the third result of our article generalises Lamzouri's results for number fields.

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“…with the exception of at most one real zero β 1 , where n L is the degree of L. It has been proven in [Kad12, Theorem 1.1] that for |Im(s)| ≤ 1, A = 12.74, and A ′ = 0 assuming d L is sufficiently large. Recently, Lee improved this to A = 12.44 in [Lee,Theorem 2]. Also, this was made explicit in [AhKw19-1, Proposition 6.1] with A = A ′ = 29.57.…”

Section: Introductionmentioning

confidence: 97%

Primes in the Chebotarev density theorem for all number fields

Kadiri¹,

Wong²

2021

Preprint

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We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let L/K be any Galois extension of number fields such that L = Q, and let C be a conjugacy class in the Galois group of L/K. We show that there exists an unramified prime p of K such that σp = C and N p ≤ d B L with B = 310. This improves the value B = 12 577 as proven by Ahn and Kwon. In comparison to previous works on the subject, we modify the weights to detect the least prime, and we use a new version of Turán's power sum method which gives a stronger Deuring-Heilbronn (zero-repulsion) phenomenon. In addition, we refine the analysis of how the location of the potential exceptional zero for ζL(s) affects the final result. We also use Fiori's numerical verification for L up to a certain discriminant height. Finally, we provide a lower bound for the number of unramified primes p of K such that σp = C.

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On an explicit zero-free region for the Dedekind zeta-function

Cited by 8 publications

References 9 publications

Explicit Interval Estimates for Prime Numbers

Explicit Interval Estimates for Prime Numbers

Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias

Primes in the Chebotarev density theorem for all number fields

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