On a sum involving the Euler function

Olivier Bordellès; Lixia Dai; Randell Heyman; Hao Pan; Igor E. Shparlinski

On a sum involving the Euler function

Number Theory 2018-10-17 v3

Authors: Olivier Bordellès , Lixia Dai , Randell Heyman , Hao Pan , Igor E. Shparlinski

Abstract

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$ , involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the reciprocals of integers.

Keywords

multiplicative functions exponential sums upper bounds

Cite

@article{arxiv.1808.00188,
  title  = {On a sum involving the Euler function},
  author = {Olivier Bordellès and Lixia Dai and Randell Heyman and Hao Pan and Igor E. Shparlinski},
  journal= {arXiv preprint arXiv:1808.00188},
  year   = {2018}
}

Comments

This version incorporate the improvement of the upper bound of Theorem 2.1 in v.1, given by Lixia Dai and Hao Pan in arXiv:1809.10381. Consecutively, Lixia Dai and Hao Pan are now co-authors

On a sum involving the Euler function

Abstract

Keywords

Cite

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