On the Behavior of Power Series with Completely Additive Coefficients

Oleg Petrushov

Bulletin of the Polish Academy of Sciences. Mathematics (2015)

  • Volume: 63, Issue: 3, page 217-225
  • ISSN: 0239-7269

Abstract

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Consider the power series ( z ) = n = 1 α ( n ) z , 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 2 π i l / q . We give effective omega-estimates for ( 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.

How to cite

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Oleg Petrushov. "On the Behavior of Power Series with Completely Additive Coefficients." Bulletin of the Polish Academy of Sciences. Mathematics 63.3 (2015): 217-225. <http://eudml.org/doc/281243>.

@article{OlegPetrushov2015,
abstract = {Consider the power series $(z) = ∑_\{n=1\}^\{∞\} α(n)zⁿ$, 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 $(e(l/p^k)r)$ 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.},
author = {Oleg Petrushov},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {power series; additive functions; growth conditions; omegaestimates; boundary behavior; arithmetic functions},
language = {eng},
number = {3},
pages = {217-225},
title = {On the Behavior of Power Series with Completely Additive Coefficients},
url = {http://eudml.org/doc/281243},
volume = {63},
year = {2015},
}

TY - JOUR
AU - Oleg Petrushov
TI - On the Behavior of Power Series with Completely Additive Coefficients
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2015
VL - 63
IS - 3
SP - 217
EP - 225
AB - Consider the power series $(z) = ∑_{n=1}^{∞} α(n)zⁿ$, 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 $(e(l/p^k)r)$ 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.
LA - eng
KW - power series; additive functions; growth conditions; omegaestimates; boundary behavior; arithmetic functions
UR - http://eudml.org/doc/281243
ER -

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