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On the behavior of power series with positive completely multiplicative coefficients. (English) Zbl 1462.11077
In a series of previous papers such as [Bull. Pol. Acad. Sci. Math. 63, 217–225 (2015); Math. Notes 103, 797–810 (2018; Zbl 1397.30003)], the author studied power series whose coefficients are values of completely multiplicative functions, and proved some asymptotic estimates as such a power series tends to the roots of \(1\) along the radii of the unit circle. This implies in particular that these series cannot be extended beyond the unit disk. The purpose of the present paper is to enlarge the set of completely multiplicative functions for which the associated power series has the unit circle as natural boundary. More precisely, it is shown that, if \(f\) is a completely multiplicative function such that
(i) For any non-principal Dirichlet character \(\chi\), the series \(\displaystyle \sum_p \frac{f(p)}{p} \left( 1 - \textrm{Re} \; \chi(p) \right)\) diverges;
(ii) the series \(\displaystyle \sum_p \frac{f(p)}{p^\sigma} \) converges for \(\sigma > 1\) ;
then, if the power series \(\displaystyle \sum_n f(n) z^n\) has a non-singular point on the unit circle, then \(f(n) \equiv 1\) identically. The proof uses \(\Omega\)-results for twisted partial sums of \(f(n)\) and the study of the twisted Dirichlet series \(\displaystyle \sum_n f(n) e(\beta n) n^{-s}\) for some \(\beta \in \mathbb{Q}\).
11N37 Asymptotic results on arithmetic functions
30B30 Boundary behavior of power series in one complex variable; over-convergence
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