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    • Fulltext

       

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/133/0001

    • Keywords

       

      Average orders; multiplicative functions.

    • Abstract

       

      When restricted to some non-negative multiplicative function, say $f$, boundedon primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n\leq X} f (n)/n$ with error term $\mathcal{O}(({\rm log} X)^{\kappa - h - 1 + \epsilon}$ (for any positive $\epsilon > 0$) as soon as we have $\sum_{p\leq Q} f (p)(\log p)/p = \kappa \log Q +\eta +O(1/(\log 2Q)^h)$ for a non-negative $\kappa$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fa\u{i}nle\u{i}b and uses a differential equation.

    • Author Affiliations

       

      OLIVIER RAMARE1 ALISA SEDUNOVA2 RITIKA SHARMA3

      1. CNRS/Institut de Mathematiques de Marseille, U.M.R. 7373, Aix Marseille Universite, Site Sud, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France
      2. Georg-August-Universitat Gottingen, Bunsenstrasse 3-5, 37073 Gottingen, Germany
      3. Stockholm University, Frescativagen, 11419 Stockholm, Sweden

    • Dates

       
      Published
      12 January 2023
      Manuscript received
      20 January 2022
      Manuscript revised
      12 September 2022
      Accepted
      29 September 2022
      Final version published online
      12 January 2023

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