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On the Behavior of Power Series with Completely Additive Coefficients

Petrushov O.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

217--225

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 e2πil/q. We give effective omega-estimates for A(e(l/pk)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.