A Cesàro average for an additive problem with prime powers

Tom 118 / 2019

Alessandro Languasco, Alessandro Zaccagnini Banach Center Publications 118 (2019), 137-152 MSC: Primary 11P32; Secondary 44A10. DOI: 10.4064/bc118-9


In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes (the paper of the authors in Forum Math. 27 (2015), see also the paper of A.L., Riv. Mat. Univ. di Parma 7 (2016), Theorem 2.2). Let $1\le \ell_1 \le \ell_2$ be two integers, $\Lambda$ be the von Mangoldt function and $r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} \Lambda(m_1) \Lambda(m_2)$ be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We prove that the Cesàro average of weight $k \gt 1$ of $r_{\ell_1,\ell_2}$ over the interval $[1, N]$ has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.


  • Alessandro LanguascoDipartimento di Matematica “Tullio Levi-Civita”
    Università di Padova
    Via Trieste 63
    35121 Padova, Italy
  • Alessandro ZaccagniniDipartimento di Scienze Matematiche, Fisiche e Informatiche
    Università di Parma
    Parco Area delle Scienze, 53/a
    43124 Parma, Italy

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