Oscillations of the remainder term related to the Euler totient function

Abstract

We split the remainder term in the asymptotic formula for the mean of the Euler phi function into two summands called the arithmetic and the analytic part respectively. We show that the arithmetic part can be studied with a mild use of the complex analytic tools, whereas the study of the analytic part heavily depends on the properties of the Riemann zeta function and on the distribution of its non-trivial zeros in particular.

Lower estimates related to the twisted Dedekind function

“…In a series of papers [2][3][4] we developed a method for proving omega results for a large number of arithmetical error terms including those related to the twisted Euler totient function or similar arithmetic functions defined using Fourier coefficients of cusp forms. Typically, lower estimates in such theorems are just by a power of the iterated logarithm sharper than trivial ones and have the following form Ω(x(log log x) θ ) or Ω ± (x(log log x) θ ) with various exponents θ ≤ 1/2.…”

Asymptotic formula for balanced words

“…Here we used Lemma 6 and the Mertens formula [22,6,24,19,13]. However in (6), the effect of this error term cancels out and we find the second main term in (7).…”

On summatory arithmetic functions and a Volterra Integral equation

“…The far right hand side of (1.9) may be recognized as the error term E(x) for the summatory Euler totient function [3], which has been estimated in [7].…”