On the normal number of prime factors of φ(n) subject to certain congruence conditions

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• 作者：M. Mkaouar<sup> ; sup><sup>mohamed.mkaouar@fss.rnu.tn" class="auth_mail" title="E-mail the corresponding authorsup> ; W. Wannes <sup>walid.wannes@fss.rnu.tn" class="auth_mail" title="E-mail the corresponding authorsup>
• 关键词：11A63 ; 11L03 ; 11N05
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：629-645
• 全文大小：328 K

文摘

Let <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si1.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=7fa4a2d6efd118cdf66054e01dee3ece" title="Click to view the MathML source">q≥2span><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">q≥n>2n>span>span>span> be an integer and <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si2.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=b29403e166e410d0a8f8017e310f71e3" title="Click to view the MathML source">S<sub>qsub>(n)span><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">sub>Sqsub>stretchy="false">(nstretchy="false">)span>span>span> denote the sum of the digits in base q of the positive integer n. It is proved that for every real number α and β   with <span id="mmlsi182" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si182.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=98844341868fbe6f6b150672c07d342f" title="Click to view the MathML source">α<βspan><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">α<βspan>span>span>,v class="formula" id="fm0010">v class="mathml"><span id="mmlsi4" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si4.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=1fcdb24d880cba795b88332ed41ba4c7">ss="imgLazyJSB inlineImage" height="116" width="383" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1500308X-si4.gif"><noscript>style="vertical-align:bottom" width="383" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X1500308X-si4.gif">noscript><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">splaystyle="true" columnspacing="0.2em">nalign="left">nder>variant="normal">limxstretchy="false">⟶+&infin;nder>⁡n>1n>x♯stretchy="true" maxsize="3.8ex" minsize="3.8ex">{n≤x:α≤vstretchy="true" maxsize="2.4ex" minsize="2.4ex">(φstretchy="false">(nstretchy="false">)stretchy="true" maxsize="2.4ex" minsize="2.4ex">)&minus;n>1n>n>2n>bsup>stretchy="false">(variant="normal">log⁡variant="normal">log⁡nstretchy="false">)n>2n>sup>n>1n>sqrt>n>3n>sqrt>bsup>stretchy="false">(variant="normal">log⁡variant="normal">log⁡nstretchy="false">)n>3n>n>2n>sup>≤βstretchy="true" maxsize="3.8ex" minsize="3.8ex">}nalign="left">space width="1em">space>=n>1n>sqrt>n>2n>πsqrt>nderover>vablelimits="false">&int;αβnderover>sup>e&minus;sup>tn>2n>sup>n>2n>sup>dt,span>span>span>ss="temp" src="/sd/blank.gif">v>v> where <span id="mmlsi184" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si184.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=690cf47a4f10c1e33b9a573464e41a48" title="Click to view the MathML source">v(n)span><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">vstretchy="false">(nstretchy="false">)span>span>span> is either <span id="mmlsi186" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si186.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=50ca68ef382491f7dee576d17b104484">ss="imgLazyJSB inlineImage" height="16" width="34" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1500308X-si186.gif"><noscript>style="vertical-align:bottom" width="34" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X1500308X-si186.gif">noscript><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">ver accent="true">ω˜ver>stretchy="false">(nstretchy="false">)span>span>span> or <span id="mmlsi185" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si185.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=86e8d6a05eca7589c7b30518f1d6f8ac">ss="imgLazyJSB inlineImage" height="20" width="35" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1500308X-si185.gif"><noscript>style="vertical-align:bottom" width="35" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X1500308X-si185.gif">noscript><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">ver accent="true">variant="normal">Ω˜ver>stretchy="false">(nstretchy="false">)span>span>span>, the number of distinct prime factors and the total number of prime factors p of a positive integer n   such that <span id="mmlsi8" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si8.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=a26fd946674fd17cdc46129b97fb0437">ss="imgLazyJSB inlineImage" height="16" width="118" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1500308X-si8.gif"><noscript>style="vertical-align:bottom" width="118" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X1500308X-si8.gif">noscript><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">sub>Sqsub>stretchy="false">(pstretchy="false">)&equiv;aspace width="0.25em">space>variant="normal">modspace width="0.25em">space>bspan>span>span> (<span id="mmlsi9" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si9.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=dd558e388887208c30062e6fc0ad289f" title="Click to view the MathML source">a,b&isin;Zspan><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">a,b&isin;variant="double-struck">Zspan>span>span>, <span id="mmlsi10" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1500308X&_mathId=si10.gif&_user=111111111&_pii=S0022314X1500308X&_rdoc=1&_issn=0022314X&md5=e522936180990377f83c7fc5bcf95296" title="Click to view the MathML source">b≥2span><span class="mathContainer hidden"><span class="mathCode">verflow="scroll">b≥n>2n>span>span>span>). This extends the results known through the work of P. Erdős and C. Pomerance, M.R. Murty and V.K. Murty to primes under digital constraint.