On the Behavior of Power Series with Completely Additive Coefficients

• Autorzy: Petrushov O.

Identyfikatory

DOI

10.4064/ba8018-1-2016

• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

power series; additive functions; growth conditions; omega-estimates; boundary behavior; arithmetic functions

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 3
• Strony: 217--225
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Petrushov O.

• Moscow State University, Moscow, Russia

Bibliografia

• [H] G. H. Hardy, On a case of term-by-term integration of an infinite series, Messenger Math. 39 (1910), 136-139.
• [L] L. G. Lucht, Power series with multiplicative coefficients, Math. Z. 177 (1981), 359-374.
• [LS] L. G. Lucht and A. Schmalmack, Polylogarithms and arithmetic function spaces, Acta Arith. 95 (2000), 361-382.
• [P1] O. A. Petrushov, On the behavior close to the unit circle of the power series with Möbius function coefficients, Acta Arith. 164 (2014), 119-136.
• [P2] O. A. Petrushov, On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values, Acta Arith. 168 (2015), 17-30.