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

Volume 164 / 2014

Oleg Petrushov Acta Arithmetica 164 (2014), 119-136 MSC: Primary 11N37; Secondary 30B30. DOI: 10.4064/aa164-2-2

Abstract

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.

Authors

  • Oleg PetrushovFaculty of Mechanics and Mathematics
    Moscow State University
    Vorobyovi Gory
    Moscow, Russia
    e-mail

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