Partial sums of biased random multiplicative functions

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• 作者：M. Aymone^a ; ^{marco@mat.ufmg.br" class="auth_mail" title="E-mail the corresponding author} ; V. Sidoravicius^b ; ^c ; ^d ; ^{vs1138@nyu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Random multiplicative functions ; Probabilistic Number Theory ; Riemann Hypothesis
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：172
• 期：Complete
• 页码：343-382
• 全文大小：675 K

文摘

Let 878fb3323686525ea1bd26347ebd" title="Click to view the MathML source">P$\mathcal{P}$ 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∈P${\{f(p)\}}_{p\in \mathcal{P}}$ form a sequence of ±1 valued independent random variables with Ef(p)<0$\mathbb{E}f(p)<0$, e9bc21807128732" title="Click to view the MathML source">∀p∈P$\forall p\in \mathcal{P}$. The function f   is called strongly biased (towards classical Möbius function), if ae529e624c34f6432">${\sum }_{p\in \mathcal{P}}\frac{f(p)}{p}=-\infty$a.s.  , and it is weakly biased if ${\sum }_{p\in \mathcal{P}}\frac{f(p)}{p}$ converges a.s.   Let e525d81ca996b97e725c" title="Click to view the MathML source">M_f(x):=∑_n≤xf(n)$M_f(x):={\sum }_{n\le x}f(n)$. We establish a number of necessary and sufficient conditions for 870cd50807971320c3b71f4d2f8" title="Click to view the MathML source">M_f(x)=o(x^1−α)$M_f(x)=o(x^{1-\alpha })$ for some 8791763024de0fc713ef6882d" title="Click to view the MathML source">α>0$\alpha >0$, a.s., when f   is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if e50cc2696ff8afbb" title="Click to view the MathML source">M_fα(x)=o(x^1/2+ϵ)$M_{f_{\alpha }}(x)=o(x^{1/2+ϵ})$ for all ϵ>0$ϵ>0$a.s.  , for each 8791763024de0fc713ef6882d" title="Click to view the MathML source">α>0$\alpha >0$, where {f_α}_α${\{f_{\alpha }\}}_{\alpha }$ is a certain family of weakly biased random multiplicative functions.