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Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski

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Abstract

Very recently Bordellès, Dai, Heyman, Pan and Shparlinski studied asymptotic behaviour of the quantity

$$\begin{aligned} S_f(x) := \sum _{n\leqslant x} f\left( \left[ \frac{x}{n}\right] \right) , \end{aligned}$$

and established some asymptotic formulas for \(S_f(x)\) under three different types of assumptions on f. In this short note we improve some of their results.

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References

  1. O. Bordellès, Short interval results for certain arithmetic functions. Int. J. Number Theory (2018). https://doi.org/10.1142/S1793042118500331

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  2. O. Bordellès, L. Dai, R. Heyman, H. Pan, I.E. Shparlinski, On a Sum Involving the Euler Function. arXiv:1808.00188v3 [math.NT], 15 Oct 2018

  3. S.W. Graham, G. Kolesnik, Van der Corput’s Method of Exponential Sums (Cambridge University Press, Cambridge, 1991)

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  4. J. Wu, On a Sum Involving the Euler Totient Function. Mathematicae 30, 536–541 (2019)

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Acknowledgements

This work is supported in part by Scientific Research Innovation Team Project Affiliated to Yangtze Normal University (No. 2016XJTD01) and NSF of Chongqing (cstc2019jcyj-msxm1651).

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Wu, J. Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski. Period Math Hung 80, 95–102 (2020). https://doi.org/10.1007/s10998-019-00300-6

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  • DOI: https://doi.org/10.1007/s10998-019-00300-6

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