Journal of the Korean Mathematical Society (대한수학회지)

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

  • Received : 2021.06.22
  • Accepted : 2022.06.28
  • Published : 2022.09.01
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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 dL ≥ d1 and x ≥ exp (326n𝛿1L(log dL)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

Acknowledgement

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(NRF-2020R1I1A1A01069868) and a Korea University Grant. The second author was supported by NRF-2019R1A2C1002786 and the College of Education, Korea University Grant.

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