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

100%
Petrushov O.
Acta Arithmetica
|
2014
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tom 164
|
nr 2
119-136
EN
Let $𝔐 (z) = ∑_{n=1}^{∞} μ(n)z^n$. We prove that for each root of unity $e(β) = e^{2πiβ}$ there is an a > 0 such that $𝔐 (e(β)r) = Ω((1-r)^{-a})$ as r → 1-. For roots of unity e(l/q) with q ≤ 100 we prove that these omega-estimates are true with a = 1/2. From omega-estimates for 𝔐 (z) we obtain omega-estimates for some finite sums.
2
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On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values

100%
Petrushov O.
Acta Arithmetica
|
2015
|
tom 168
|
nr 1
17-30
EN
We consider the behavior of the power series $𝔐_0(z) = ∑_{n=1}^{∞} μ^2(n)z^n$ as z tends to $e(β) = e^{2πiβ}$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then $𝔐_0(e(β)r) = O((1-r)^{-1/2-ε})$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that $𝔐_0(e(β)r) = Ω((1-r)^{-1+δ})$.
3
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On the Behavior of Power Series with Completely Additive Coefficients

100%
Petrushov O.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
|
tom 63
|
nr 3
217-225
EN
Consider the power series $𝔄(z) = ∑_{n=1}^{∞} α(n)zⁿ$, where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity $e^{2πil/q}$. We give effective omega-estimates for $𝔄(e(l/p^k)r)$ when r → 1-. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.
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