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

Oleg Petrushov. "On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values." Acta Arithmetica 168.1 (2015): 17-30. <http://eudml.org/doc/279033>.

@article{OlegPetrushov2015,
abstract = {We consider the behavior of the power series $_0(z) = ∑_\{n=1\}^\{∞\} μ^2(n)z^n$ as z tends to $e(β) = e^\{2πiβ\}$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then $_0(e(β)r) = O((1-r)^\{-1/2-ε\})$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that $_0(e(β)r) = Ω((1-r)^\{-1+δ\})$.},
author = {Oleg Petrushov},
journal = {Acta Arithmetica},
keywords = {power series; squared Möbius function; omega-estimates; arithmetic functions; irrationality exponent},
language = {eng},
number = {1},
pages = {17-30},
title = {On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values},
url = {http://eudml.org/doc/279033},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Oleg Petrushov
TI - On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 1
SP - 17
EP - 30
AB - We consider the behavior of the power series $_0(z) = ∑_{n=1}^{∞} μ^2(n)z^n$ as z tends to $e(β) = e^{2πiβ}$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then $_0(e(β)r) = O((1-r)^{-1/2-ε})$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that $_0(e(β)r) = Ω((1-r)^{-1+δ})$.
LA - eng
KW - power series; squared Möbius function; omega-estimates; arithmetic functions; irrationality exponent
UR - http://eudml.org/doc/279033
ER -