On the zero distribution of the Riesz transforms of power series (Q1842730)

There is a classical result by \textit{R. Jentzsch} [Acta Math. 41, 219-251 (1917; Jahrb. F.d.M. 45, 647-649)] on convergent power series with finite radius of convergence that specifies how the zeros of its partial sums increase in number. Regarding their regular Nörlund transforms instead, the author and \textit{W. Luh} [Acta Sci. Math. 58, No. 1-4, 243-251 (1993; Zbl 0789.40004)] furnished a generalization, and a refinement as well, of that result. In the present paper, the place of \((N, p_ n)\) is taken by the Riesz weighted means method \((\overline N, p_ n)\) such that \(q : = \limsup \root n\of{| p_ n/P_ n |}< 1\) \(P_ n : = \sum^ n_{\nu = 0} p_ \nu\) (thus exluding, e.g., the method \((C,1)\) this time). Let the series \(\sum a_ \nu z^ \nu\) have radius of convergence 1, let its partial sums be transformed into the sequence \(\sigma_ n (z)\), and let \(A_ n (R) : = \# \{z : \sigma_ n (z) = 0, | z | \leq R\}\), \(R > 0\). Matching the Jentzsch result, \(\limsup {1 \over n} A_ n (R) = 1\) was proved to hold for all \(R > 1\) in case of the Nörlund transform \(\sigma_ n (z)\); with the Riesz transform \(\sigma_ n (z)\) it is true for all \(R > {1 \over q}\) provided that \(q = \lim\root n\of{| p_ n/P_ n |}\) (the proof being reported in detail to the reviewer). Yet, the main objective of both papers is to characterize the class of series such that \(\liminf {1 \over n} A_ n (R) < 1\) holds for some \(R > 1\) resp. \(R > {1 \over q} (>1)\). This was achieved by some kind of Ostrowski gap conditions on the series, involving \(n_ k < m_{k + 1}\), \(n_ k/m_ k \geq \lambda > 1\), and \(\limsup \{\alpha_ \nu : m_ k < \nu \leq n_ k\), \(k \in \mathbb{N}\} \leq \theta < 1\) with \(\alpha_ \nu = \root \nu\of {| a_ \nu |}\) in the Nörlund, \(\begin{smallmatrix} \alpha_ \nu = \left | a_ \nu \sum^{n_ k}_{\mu = \nu} p_ \mu \left / P_{n_ k} \right. \right | \end{smallmatrix}\) in the Riesz case. Furthermore, overconvergence of the Riesz transform on some \(\{z : | z | < {1 \over q} \}\) is shown to imply the condition in question.