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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=f68a878fb3323686525ea1bd26347ebd" title="Click to view the MathML source">Pclass="mathContainer hidden">class="mathCode">$\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions f   supported on the squarefree integers, such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=3bf3c73403017cc6703dcd0ec18a10a7" title="Click to view the MathML source">{f(p)}_p∈Pclass="mathContainer hidden">class="mathCode">${\{f(p)\}}_{p\in \mathcal{P}}$ form a sequence of ±1 valued independent random variables with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=a79e6d68f810d84d4ecea3d51aaa8026" title="Click to view the MathML source">Ef(p)<0class="mathContainer hidden">class="mathCode">$\mathbb{E}f(p)<0$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si21.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=3756b0d44f5fcfbf0e9bc21807128732" title="Click to view the MathML source">∀p∈Pclass="mathContainer hidden">class="mathCode">$\forall p\in \mathcal{P}$. The function f   is called strongly biased (towards classical Möbius function), if class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si22.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=98cad11178ab540ae529e624c34f6432">class="imgLazyJSB inlineImage" height="24" width="126" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302347-si22.gif">class="mathContainer hidden">class="mathCode">${\sum }_{p\in \mathcal{P}}\frac{f(p)}{p}=-\infty$a.s.  , and it is weakly biased if class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=d1e289814a8d8605515d374e00c48913">class="imgLazyJSB inlineImage" height="24" width="75" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302347-si6.gif">class="mathContainer hidden">class="mathCode">${\sum }_{p\in \mathcal{P}}\frac{f(p)}{p}$ converges a.s.   Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si7.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=9e62eb7abe73e525d81ca996b97e725c" title="Click to view the MathML source">M_f(x):=∑_n≤xf(n)class="mathContainer hidden">class="mathCode">$M_f(x):={\sum }_{n\le x}f(n)$. We establish a number of necessary and sufficient conditions for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si79.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=71146870cd50807971320c3b71f4d2f8" title="Click to view the MathML source">M_f(x)=o(x^1−α)class="mathContainer hidden">class="mathCode">$M_f(x)=o(x^{1-\alpha })$ for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si35.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=55ea7648791763024de0fc713ef6882d" title="Click to view the MathML source">α>0class="mathContainer hidden">class="mathCode">$\alpha >0$, a.s., when f   is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si10.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=81df3e336a1b4d7ee50cc2696ff8afbb" title="Click to view the MathML source">M_fα(x)=o(x^1/2+ϵ)class="mathContainer hidden">class="mathCode">$M_{f_{\alpha }}(x)=o(x^{1/2+ϵ})$ for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si11.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=9fac55dc03ca202f509f6e140c44ead5" title="Click to view the MathML source">ϵ>0class="mathContainer hidden">class="mathCode">$ϵ>0$a.s.  , for each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si35.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=55ea7648791763024de0fc713ef6882d" title="Click to view the MathML source">α>0class="mathContainer hidden">class="mathCode">$\alpha >0$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302347&_mathId=si12.gif&_user=111111111&_pii=S0022314X16302347&_rdoc=1&_issn=0022314X&md5=4383aa3321939805da4bf09c8e701c25" title="Click to view the MathML source">{f_α}_αclass="mathContainer hidden">class="mathCode">${\{f_{\alpha }\}}_{\alpha }$ is a certain family of weakly biased random multiplicative functions.