Eine Bemerkung über die Werte der Funktion σ (n)

Abstract

By means of the Hoheisel—Montgomery prime number theorem it is shown that for every α≥1 the inequality

$$|(\sigma (n)/n) - \alpha | \leqslant {1 \mathord{\left/ {\vphantom {1 {n^{({2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-\nulldelimiterspace} 5}) - \varepsilon } }}} \right. \kern-\nulldelimiterspace} {n^{({2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-\nulldelimiterspace} 5}) - \varepsilon } }}(\varepsilon > 0,\sigma (n) = \sum\limits_{d/n} d )$$

has infinitely many solutionsn∈N. It is highly probable that the exponent 2/5 can be replaced by 1.