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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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