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Partial sums of biased random multiplicative functions. (English) Zbl 1414.11115
Summary: Let \(\mathcal{P}\) be the set of the primes. We consider a class of random multiplicative functions \(f\) supported on the squarefree integers, such that \(\{f(p) \}_{p \in \mathcal{P}}\) form a sequence of \(\pm\)1 valued independent random variables with \(\mathbb{E} f(p) < 0\), \(\text{for all } p \in \mathcal{P}\). The function \(f\) is called strongly biased (towards classical Möbius function), if \(\sum_{p \in \mathcal{P}} \frac{f(p)}{p} = - \infty\) a.s., and it is weakly biased if \(\sum_{p \in \mathcal{P}} \frac{f(p)}{p}\) converges a.s. Let \(M_f(x) : = \sum_{n \leq x} f(n)\). We establish a number of necessary and sufficient conditions for \(M_f(x) = o(x^{1 - \alpha})\) for some \(\alpha > 0\), a.s., when \(f\) is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if \(M_{f_\alpha}(x) = o(x^{1 / 2 + \varepsilon})\) for all \(\varepsilon > 0\) a.s., for each \(\alpha > 0\), where \(\{f_\alpha \}_\alpha\) is a certain family of weakly biased random multiplicative functions.
MSC:
11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
11K65 Arithmetic functions in probabilistic number theory
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