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On a sum involving certain arithmetic functions and the integral part function

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Abstract

In this short paper, we study sums of the shape \(\sum _{n\leqslant x}{f([x/n])}/{[x/n]},\) where f is the Euler totient function \(\varphi \), the Dedekind function \(\Psi \), the sum-of-divisors function \(\sigma \) or the alternating sum-of-divisors function \(\beta .\) We improve previous results when \(f=\varphi \) and derive new estimates when \(f=\Psi , f=\sigma \) and \(f=\beta .\)

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Acknowledgements

The authors are grateful for Professor Jie Wu’s instructive talk and discussion.

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  1. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Jing Ma & Huayan Sun

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  1. Jing Ma
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  2. Huayan Sun
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Correspondence to Huayan Sun.

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Ma, J., Sun, H. On a sum involving certain arithmetic functions and the integral part function. Ramanujan J 60, 1025–1032 (2023). https://doi.org/10.1007/s11139-022-00588-y

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  • DOI: https://doi.org/10.1007/s11139-022-00588-y

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