Exponentialsummen mit der Möbiusfunktion

Abstract

The study of sums that contain the Möbius- function has a long tradition as we have already indicated. The aim of this work was to estimate such sums, in which Dirichlet-characters modulo q occur as well and the sum runs only over those numbers that do not contain large prime factors.

The summation could be reduced by Perron"s formula to an integral, while two mean value calculations were carried out, one over the imaginary parts and one over the Dirichlet- characters.

Claudia Fischer has already used in her thesis an averaging over the imaginary parts. The averaging over the Dirichlet- characters was made because the path of integration is chosen specifically for each Dirichlet- character.

Perron"s formula leads us to an integral with two parts. The first part is calculated based on the method of Baker and Harman combined with the path of integration of Maier and Montgomery as a piecewise linear contour. Here, we use monotonicity principles on horizontal lines which are parallel to the real axis and it is examined how many times the value of the inverse of the Dirichlet- L- series exceeds a certain limit. This is determined by dividing the candidate pairs of imaginary parts and characters into three sets and estimating their contribution. In the second part we use the inclusion- exclusion principle to estimate uniformly the sum contained in the integral.