Abstract
When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for \(\sum _{n\le X}f(n)/n\) with error term \({\mathcal {O}}((\log X)^{\kappa -h-1+\varepsilon })\) (for any positive \(\varepsilon >0\)) as soon as we have \(\sum _{p\le Q}f(p)(\log p)/p=\kappa \log Q+\eta +{\mathcal {O}}(1/(\log 2Q)^h)\) for a non-negative \(\kappa \) and some non-negative integer h. The method generalizes the 1967-approach of Levin and Faĭnleĭb and uses a differential equation.
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Acknowledgements
This paper started in 2018 when the first and third authors were invited by the Indian Statistical Institute, Delhi under Cefipra program 5401-A. It was continued when these authors were visiting Stockholm in early 2019 and then in July of the same year when the first and second authors were invited by the Max Planck Institute in Bonn. It was finalized in 2021 when the first author was invited by the Haussdorf Institut für Mathematik in Bonn and the second author was invited by the Max Planck Institute in Bonn. The authors would like to thank all these institutions for providing suitable conditions without which this piece of work would surely have died in their drawers.
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Ramaré, O., Sedunova, A. & Sharma, R. A higher order Levin–Faĭnleĭb theorem. Proc Math Sci 133, 1 (2023). https://doi.org/10.1007/s12044-022-00721-3
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DOI: https://doi.org/10.1007/s12044-022-00721-3