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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cesàro average in short intervals for Goldbach numbers
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by Alessandro Languasco and Alessandro Zaccagnini PDF
Proc. Amer. Math. Soc. 145 (2017), 4175-4186 Request permission

Abstract:

Let $\Lambda$ be the von Mangoldt function and \begin{equation*} R(n) = \sum _{h+k=n} \Lambda (h)\Lambda (k). \end{equation*} Let further $N,H$ be two integers, $N\ge 2$, $1\le H \le N$, and assume that the Riemann Hypothesis holds. Then \begin{align*} \sum _{n=N-H}^{N+H} R(n) \Bigl (1- \frac {\vert n- N \vert }{H}\Bigr ) &= HN -\frac {2}{H} \sum _{\rho } \frac {(N+H)^{\rho +2} - 2 N^{\rho +2} +(N-H)^{\rho +2} }{\rho (\rho + 1)(\rho + 2)} \\& + \mathcal {O}\Bigl ({N \Bigl (\log \frac {2N}{H}\Bigr )^2 + H (\log N)^2 \log (2H) }\ , \end{align*} where $\rho =1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta (s)$.
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Additional Information
  • Alessandro Languasco
  • Affiliation: Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, 35121 Padova, Italy
  • MR Author ID: 354780
  • ORCID: 0000-0003-2723-554X
  • Email: languasco@math.unipd.it
  • Alessandro Zaccagnini
  • Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, 43124 Parma, Italy
  • Email: alessandro.zaccagnini@unipr.it
  • Received by editor(s): June 1, 2016
  • Received by editor(s) in revised form: November 1, 2016
  • Published electronically: May 4, 2017
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4175-4186
  • MSC (2010): Primary 11P32; Secondary 11P55
  • DOI: https://doi.org/10.1090/proc/13645
  • MathSciNet review: 3690604