Limit[Log[LCM @@ (h + Range[n] k)]/n, n -> Infinity] == (k/EulerPhi[k]) Plus @@ (1/Select[Range[k], Function[m, GCD[m, k] == 1]]) /; Element[h, Integers] && h > 0 && Element[k, Integers] && Element[k > 0 n, Integers] && n > 0 && GCD[h, k] == 1

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> h </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mi> h </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> h </mi> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> k </mi> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <munder> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </munder> <mi> k </mi> </munderover> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> lim </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#62754; </ms> <ms> &#8734; </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> lcm </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> h </ms> <ms> + </ms> <ms> k </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> h </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> k </ms> </list> </apply> </list> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> h </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> k </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> n </ms> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#981; </ms> <ci> EulerPhi </ci> </apply> <ms> ( </ms> <ms> k </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> gcd </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <ms> k </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> k </ms> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> m </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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2002-12-18