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

In many cases known methods of detecting oscillations of arithmetic error terms involve certain smoothing pro‐cedures. 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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)