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NOTE ON SUMS INVOLVING THE EULER FUNCTION

  • SHANE CHERN (a1)

Abstract

In this note, we provide refined estimates of two sums involving the Euler totient function,

$$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$
where $[x]$ denotes the integral part of real $x$ . The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].

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[1] Apostol, T. M., Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics (Springer, New York–Heidelberg, 1976).
[2] Bordellès, O., Arithmetic Tales, translated from the French by Véronique Bordellès, Universitext (Springer, London, 2012).
[3] Bordellès, O., Dai, L., Heyman, R., Pan, H. and Shparlinski, I. E., ‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188.10.1016/j.jnt.2019.01.006
[4] Hardy, G. H., ‘On Dirichlet’s divisor problem’, Proc. Lond. Math. Soc. (2) 15 (1916), 125.
[5] Huxley, M. N., ‘Exponential sums and lattice points. III’, Proc. Lond. Math. Soc. (3) 87(3) (2003), 591609.
[6] Vaaler, J. D., ‘Some extremal functions in Fourier analysis’, Bull. Amer. Math. Soc. (N.S.) 12(2) (1985), 183216.
[7] Wu, J., ‘On a sum involving the Euler totient function’, Preprint, 2018, available at hal-01884018.
[8] Wu, J., ‘Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski’, Period. Math. Hungar., to appear.
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NOTE ON SUMS INVOLVING THE EULER FUNCTION

  • SHANE CHERN (a1)

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