On the Behavior of Power Series with Completely Additive Coefficients

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 -