Partial sums of biased random multiplicative functions

Abstract

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 $\mathbb{E}f(p)<0$, $\forall p\in \mathcal{P}$. 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 $M_f(x):={\sum }_{n\le x}f(n)$. 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 $M_{f_{\alpha }}(x)=o(x^{1/2+\epsilon })$ for all $\epsilon >0$ a.s., for each $\alpha >0$, where $\{f_{\alpha }{\}}_{\alpha }$ is a certain family of weakly biased random multiplicative functions.

29 pages, Corrected typos, new section with concluding remarks

References in corpus (2)

• Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function
• On multiplicative functions which are small on average

Cited by in corpus (1)

• On multiplicative functions which are small on average and zero free regions for the Riemann zeta function