On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values

Oleg Petrushov¹

¹ Moscow State University Vorobyovy Gory Moscow, Russia

Acta Arithmetica, Tome 168 (2015) no. 1, pp. 17-30

Résumé

We consider the behavior of the power series $\mathfrak{M}_0(z)=\sum_{n=1}^{\infty}\mu^2(n)z^n$ as $z$ tends to $e(\beta)=e^{2\pi i\beta}$ along a radius of the unit circle. If $\beta$ is irrational with irrationality exponent 2 then $\mathfrak{M}_0(e(\beta)r)=O((1-r)^{-1/2-\varepsilon})$. Also we consider the cases of higher irrationality exponent. We prove that for each $\delta$ there exist irrational numbers $\beta$ such that $\mathfrak{M}_0(e(\beta)r)=\Omega((1-r)^{-1+\delta})$.

Détail

DOI : 10.4064/aa168-1-2

Keywords: consider behavior power series mathfrak sum infty tends beta beta along radius unit circle beta irrational irrationality exponent mathfrak beta r varepsilon consider cases higher irrationality exponent prove each delta there exist irrational numbers beta mathfrak beta omega r delta