On the Sum of the Twisted Euler Function

Abstract

We study an asymptotic formula for the sum of values of the Euler φ-function twisted by a real Dirichlet character. The error term is split into the arithmetic and the analytic part. The former is studied with minimal use of analytic tools in contrast to the latter, where the analysis depends heavily on the distribution of the non-trivial zeros of the corresponding Dirichlet L-function. The results of the present paper are an extension of a recent work by the authors, where the case of the classical Euler φ-function has been studied. The present, more general situation invites new technical difficulties. Not all of them can be successfully overcome. For instance, satisfactory omega results for the analytic part are proved in the case of an even Dirichlet character only. Nevertheless, a method providing good omega estimates for the arithmetic part as well as for the complete error term is developed. Moreover, it is noted that the Riemann Hypothesis for the involved Dirichlet L-function is equivalent to a sufficiently sharp estimation of the analytic part. This shows in particular that the arithmetic part can be much larger than the corresponding analytic part.

On the solution of the Volterra integral equation of second type for the error term in an asymptotic formula for arithmetical functions

“…2.Related considerations for the twisted Euler ϕ-function J.Kaczorowski and K.Wiertelak obtained a better decomposition for the remainder term in the asymptotic formula for a generalization of the Euler totient function (see [3]) : For a non-principal real Dirichlet character χ (mod q), q > 2, let ϕ(n, χ) denote the twisted Euler ϕ-function…”

“…where A is an arbitrary constant. (In the paper [3], the unique solution is F(x, χ) = x f (x, χ), but the comments just after (1.4) should also be applied here.) Moreover, when A = 0 in (2.6), for x ≥ 0…”

Zero-Free Primitive Square Representations by Some Ternary Quadratic Forms

“…These kinds of sums have been recently studied by Kaczorowski [17] and Kaczorowski and Wiertelak [18,19]. In our context, one must additionally account for a separation between sums over odd and even numbers.…”

On the Volterra integral equation for the remainder term in the asymptotic formula on the associated Euler totient function

“…1. Introduction J.Kaczorowski and K.Wiertelak obtained a decomposition for the remainder term in the asymptotic formula for a generalization of the Euler totient function (see [3]) : For a non-principal real Dirichlet character χ (mod q), q > 2, let ϕ(n, χ) denote the twisted Euler ϕ-function…”

“…Theorem 1.1 (Theorem 1.1 in [3]). The solution of the following Volterra integral equation of second type…”