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http://dx.doi.org/10.4134/JKMS.j210393

A GENERALIZATION OF MAYNARD'S RESULTS ON THE BRUN-TITCHMARSH THEOREM TO NUMBER FIELDS

Ahn, Jeoung-Hwan (Institute of Basic Science Korea University)
Kwon, Soun-Hi (Department of Mathematics Education Korea University)

Publication Information

Journal of the Korean Mathematical Society / v.59, no.5, 2022 , pp. 843-867 More about this Journal

Abstract

Maynard proved that there exists an effectively computable constant q₁ such that if q ≥ q₁, then $\frac{{\log}\;q}{\sqrt{q}{\phi}(q)}Li(x){\ll}{\pi}(x;\;q,\;m)<\frac{2}{{\phi}(q)}Li(x)$ for x ≥ q⁸. In this paper, we will show the following. Let 𝛿₁ and 𝛿₂ be positive constants with 0 < 𝛿₁, 𝛿₂ < 1 and 𝛿₁ + 𝛿₂ > 1. Assume that L ≠ ℚ is a number field. Then there exist effectively computable constants c₀ and d₁ such that for d_L ≥ d₁ and x ≥ exp (326n^𝛿₁_L(log d_L)^1+𝛿₂), we have $\|{\pi}_C(x)-\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)\|\;{\leq}\;$1-c_0\frac{1og\;d_L}{d^{7.072}_L}$\;\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)$ .

Keywords

The Chebotarev density theorem; the Deuring-Heilbronn phenomenon; the Brun-Titchmarsh theorem;

Citations & Related Records

Reference

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