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On a Smoothed Average of the Number of Goldbach Representations

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Number Theory in Memory of Eduard Wirsing

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Abstract

Assuming the Generalized Riemann Hypothesis for the zeros of the Dirichlet L-functions with characters modulo q, we obtain a smoothed version of the average number of Goldbach representations for numbers which are multiples of a positive integer q. Such an average was first considered by Granville [11, 12] but without any smoothing factor. In this short article, we also show how the smoothing can be removed.

Dedicated to the memory of Eduard Wirsing

The second author was supported by JSPS KAKENHI Grant Numbers 18K13400 and 22K13895, and also by MEXT Initiative for Realizing Diversity in the Research Environment.

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Notes

  1. 1.

    We have \(G_q(N)=0\) if \(q>N\).

  2. 2.

    If \(\chi \) in (28) is imprimitive one can still bound \(J_1\) and \(J_2\) by replacing \(\chi \) with the primitive character \(\chi ^*\) which induces it in the sums in the integrands of (28) and add an insignificant error term of \(O(\log N\log q)\) to the resulting sums.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, San Jose State University, San Jose, CA, USA

    Daniel A. Goldston

  2. Faculty of Mathematics, Kyushu University, Fukuoka, Japan

    Ade Irma Suriajaya

Authors
  1. Daniel A. Goldston
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  2. Ade Irma Suriajaya
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Correspondence to Ade Irma Suriajaya .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Ulm, Ulm, Germany

    Helmut Maier

  2. Institut für Mathematik, University of Würzburg, Würzburg, Germany

    Jörn Steuding

  3. Institut für Mathematik, University of Würzburg, Würzburg, Germany

    Rasa Steuding

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Goldston, D.A., Suriajaya, A.I. (2023). On a Smoothed Average of the Number of Goldbach Representations. In: Maier, H., Steuding, J., Steuding, R. (eds) Number Theory in Memory of Eduard Wirsing. Springer, Cham. https://doi.org/10.1007/978-3-031-31617-3_10

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