Abstract
An upper bound estimate in the law of the iterated logarithm for Σf(n k ω) where nk+1∫nk≧ 1 + ck -α (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n k} is assumed. As an application, the law of the iterated logarithm is proved when {n k} is the arrangement in increasing order of the set B(τ)={1 i 1...qτ i τ|i1,...,iτ∈N 0}, where τ≧ 2, N 0=NU{0}, and q 1,...,q τ are integers greater than 1 and relatively prime to each others.
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Fukuyama, K., Petit, B. An asymptotic property of gap series. II. Acta Mathematica Hungarica 98, 245–258 (2002). https://doi.org/10.1023/A:1022877909906
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DOI: https://doi.org/10.1023/A:1022877909906