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A singular series average and the zeros of the Riemann zeta-function

Volume 200 / 2021

D. A. Goldston, Ade Irma Suriajaya Acta Arithmetica 200 (2021), 71-90 MSC: 11N05, 11M26. DOI: 10.4064/aa200821-24-2 Published online: 28 June 2021


We show that the Riesz mean of the singular series in the Goldbach and the Hardy–Littlewood Prime Pair Conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the Riemann zeta-function. Unconditionally this error term can be shown to oscillate, while conditionally it can be shown to oscillate between sharp bounds.


  • D. A. GoldstonDepartment of Mathematics and Statistics
    San José State University
    1 Washington Square
    San José, CA 95192-0103, U.S.A.
  • Ade Irma SuriajayaFaculty of Mathematics
    Kyushu University
    744 Motooka, Nishi-ku
    Fukuoka 819-0395, Japan

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