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On some applications of probability to analysis and number theory. (English) Zbl 0125.08602
The author discusses applications of probability theory for five problems of analysis, among them the following:
1. For what sequence of integers \(n_1 < n_2 < \cdots\) does there exist a power series \(\sum_{k=1}^\infty a_k z^{n_k}\) converging uniformly in \(|z| \leq 1\) but for which \(\sum_{k=1}^\infty |a_k| = \infty?\)
2. It is known that \(f_t(z) = \sum_{k=0}^\infty \varepsilon_ka_k z^k\) where \(\varepsilon_k = \pm 1\), \(t = \sum_{k=1}^\infty {1+\varepsilon_k \over 2^{k+1}}\) and \(\sum_{k=1}^\infty |a_k|^2 = \infty\), diverges almost everywhere on the unit circle if \(|a_k| \geq c_k\) where \(c_k > 0\) is a monotone sequence of numbers tending to zero so that \[ \limsup_{k=\infty} \left[\left(\sum_{j=1}^k c_j^2\right)/\log (1/c_k) \right] > 0. \] If this does not hold, is there a sequence \(\{a_k\}\) such that \(|a_k| \geq c_k\), for which \(f_t(z)\) has at least one point of convergence for all \(t\)?
Some unpublished probabilistic methods in number theory conclude the paper.
Reviewer: J.M.Gani

MSC:
11N25 Distribution of integers with specified multiplicative constraints
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
30B10 Power series (including lacunary series) in one complex variable
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