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 mathmlsrc">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">PmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">mathvariant="script">Pmath> be the set of the primes. We consider a class of random multiplicative functions f   supported on the squarefree integers, such that mathmlsrc">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∈PmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">{f(p)}p∈mathvariant="script">Pmath> form a sequence of ±1 valued independent random variables with mathmlsrc">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)<0mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">mathvariant="double-struck">Ef(p)<0math>, mathmlsrc">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∈PmathContainer hidden">mathCode"><math altimg="si21.gif" overflow="scroll">∀p∈mathvariant="script">Pmath>. The function f   is called strongly biased (towards classical Möbius function), if mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si22.gif" overflow="scroll">∑p∈mathvariant="script">Pf(p)p=−∞math>a.s.  , and it is weakly biased if mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll">∑p∈mathvariant="script">Pf(p)pmath> converges a.s.   Let mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll">Mf(x):=∑n≤xf(n)math>. We establish a number of necessary and sufficient conditions for mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si79.gif" overflow="scroll">Mf(x)=o(x1−α)math> for some mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si35.gif" overflow="scroll">α>0math>, a.s., when f   is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll">Mfα(x)=o(x1/2+ϵ)math> for all mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll">ϵ>0math>a.s.  , for each mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si35.gif" overflow="scroll">α>0math>, where mathmlsrc">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}_αmathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll">{fα}αmath> is a certain family of weakly biased random multiplicative functions.