Smoothing arithmetic error terms: the case of the Euler φ function

Abstract

In many cases known methods of detecting oscillations of arithmetic error terms involve certain smoothing procedures. Usually an application of the smoothing operator does not change significantly the order of magnitude of the error under consideration. This is so for instance in the case of the classical error terms known in the prime number theory. The main purpose of this paper is to show that the situation for primes is not general. Considering the error term in the asymptotic formula for the Euler totient function we show that just one application of an integral smoothing operator changes situation dramatically: the order of magnitude of drops from x to √ x.

Oscillations of the remainder term related to the Euler totient function

“…Analogous result for R 1 (x) in the place of R 1 (x) + R 2 (x) was established in [12]. The present lemma follows by repeating all steps in the proof of Theorem 1.1 in [12]. The required modifications are straightforward and shall not be described here in details.…”

“…The best omega result for E(x) belongs to H.L. Montgomery [14]: was studied by the present authors in [12]. It was proved that for x tending to infinity we havẽ …”

“…We estimate these integrals exactly in the same way as in the proof of Theorem 1.2 in [12] getting (1.25). Hence (1) implies (2).…”

Products of Farey Fractions

“…In 2010 Kaczorowski and Wirtelak [9], [10] studied in more detail the oscillatory nature of the remainder term E(x). These papers show that E(x) can be split as a sum of two natural parts, an arithmetic part and an analytic part, with the analytic part having a direct connection to the zeta zeros.…”

The probability that two random integers are coprime

“…Notice that, under this distribution, the two random variables and have the same distribution but are not independent anymore. Simultaneously, they iterated this smoothing procedure in [4], and smartly recovered Montgomery's result.…”