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

Volume 168 / 2015

Oleg Petrushov Acta Arithmetica 168 (2015), 17-30 MSC: Primary 11N37; Secondary 30B30. DOI: 10.4064/aa168-1-2

Abstract

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 $\mathfrak{M}_0(e(\beta)r)=\Omega((1-r)^{-1+\delta})$.

Authors

  • Oleg PetrushovMoscow State University
    Vorobyovy Gory
    Moscow, Russia
    e-mail

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