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Zeta sum and big Oh

Source: Miklós Schweitzer 2014, P4

December 23, 2014
real analysisreal analysis unsolved

Problem Statement

For a positive integer nn, define f(n)f(n) to be the number of sequences (a1,a2,…,ak)(a_1,a_2,\dots,a_k) such that a1a2⋯ak=na_1a_2\cdots a_k=n where ai≥2a_i\geq 2 and k≥0k\ge 0 is arbitrary. Also we define f(1)=1f(1)=1. Now let α>1\alpha>1 be the unique real number satisfying ζ(α)=2\zeta(\alpha)=2, i.e ∑n=1∞1nα=2 \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 Prove that
(a) ∑j=1nf(j)=O(nα) \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) (b) There is no real number β<α\beta<\alpha such that ∑j=1nf(j)=O(nβ) \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta)
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