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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Some unpublished probabilistic methods in number theory conclude the paper.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
Show Scanned Page ##### 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
##### Keywords:
probability theory
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