On the behaviour close to the unit circle of the power series with Möbius function coefficients

Oleg Petrushov¹

¹ Faculty of Mechanics and Mathematics Moscow State University Vorobyovi Gory Moscow, Russia

Acta Arithmetica, Tome 164 (2014) no. 2, pp. 119-136.

Résumé

Let $\mathfrak {M}(z)=\sum _{n=1}^{\infty }\mu (n)z^n$. We prove that for each root of unity $e(\beta )=e^{2\pi i\beta }$ there is an $a>0$ such that $\mathfrak {M}(e(\beta )r)=\varOmega ((1-r)^{-a})$ as $r\to 1-.$ For roots of unity $e(l/q)$ with $q\le 100$ we prove that these omega-estimates are true with $a=1/2$. From omega-estimates for $\mathfrak {M}(z)$ we obtain omega-estimates for some finite sums.

Détail

DOI : 10.4064/aa164-2-2

Keywords: mathfrak sum infty prove each root unity beta beta there mathfrak beta varomega r a roots unity prove these omega estimates omega estimates mathfrak obtain omega estimates finite sums