On a sum involving the Euler function

Abstract

We obtain reasonably tight upper and lower bounds on the sum n x ϕ (⌊x/n⌋), involving the Euler functions ϕ and the integer parts ⌊x/n⌋ of the reciprocals of integers.

Note on Sums Involving the Euler Function

“…Let [x] denote the integral part of a real number x. In a recent paper, Bordellès et al [3] studied the asymptotic behaviour of the function…”

On a sum involving the Euler totient function

“…For comparaison, we have As indicated by Bordellès, Heyman and Shparlinski in [1], the proof of (1.2) is rather elementary. But the lower bound (1.3) uses a much deeper approach relying on the theory of exponential pairs.…”

“…2016XJD01). The author is thankful to Lixia Dai at Nanjing Normal University (China) for drawing his attention to Bordellès, Heyman and Shparlinski's paper [1].…”

“…van der Corput inequality. (b) Our argument about upper bound part is different from [1]. In order to improve upper bound (1.2), we introduced exponential sum technique to show that the constant in the upper bound can be improved to 1 − θ + θ 6 π 2 .…”

“…As usual, denote by ϕ(n) the Euler totient function and by [t] the integral part of real t. Very recently, Bordellès, Heyman and Shparlinski [1] studied the asymptotic behaviour of the summatory function They also posed a question : Is it true that S(x) = 6 π 2 + o(1) x log x as x → ∞ ? Some numerical evidences were given.…”