Search: id:A002321
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%I A002321 M0102 N0038
%S A002321 1,0,1,1,2,1,2,2,2,1,2,2,3,2,1,1,2,2,3,3,2,1,2,2,2,1,1,
%T A002321 1,2,3,4,4,3,2,1,1,2,1,0,0,1,2,3,3,3,2,3,3,3,3,2,2,3,3,
%U A002321 2,2,1,0,1,1,2,1,1,1,0,1,2,2,1,2,3,3,4,3,3,3,2,3,4,4,4
%V A002321 1,0,-1,-1,-2,-1,-2,-2,-2,-1,-2,-2,-3,-2,-1,-1,-2,-2,-3,-3,-2,-1,-2,-2,
               -2,-1,-1,
%W A002321 -1,-2,-3,-4,-4,-3,-2,-1,-1,-2,-1,0,0,-1,-2,-3,-3,-3,-2,-3,-3,-3,-3,-2,
               -2,-3,-3,
%X A002321 -2,-2,-1,0,-1,-1,-2,-1,-1,-1,0,-1,-2,-2,-1,-2,-3,-3,-4,-3,-3,-3,-2,-3,
               -4,-4,-4
%N A002321 Mertens's function: Sum_{1<=k<=n} mu(k), where mu = Moebius function 
               (A008683).
%C A002321 Partial sums of the Moebius function A008683.
%C A002321 Also determinant of n X n (0,1) matrix defined by A(i,j)=1 if j=1 or 
               i divides j.
%C A002321 Equals row sums of triangle A152901 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 14 2008]
%C A002321 Permanent of n X n (-1,0,1) matrix A(i,j) defined by: if j=1 or i=j then 
               A(i,j)=1 elseif i divides j then A(i,j) = -1 else A(i,j)=0. [From 
               Mats Granvik (mats.granvik(AT)abo.fi), Jul 19 2009]
%D A002321 J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 
               3, March 2003), 341-353. See p. 347.
%D A002321 Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius 
               function. Experiment. Math. 5 (1996), no. 4, 291-295.
%D A002321 E. Landau, Vorlesungen ueber Zahlentheorie, Chelsea, NY, Vol. 2, p. 157.
%D A002321 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 
               105, National Research Council, Washington, DC, 1941, pp. 7-10.
%D A002321 N. C. Ng, The summatory function of the Mobius function, Abstracts Amer. 
               Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-316.
%D A002321 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.1.
%D A002321 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002321 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002321 R. D. von Sterneck, Empirische Untersuchung ueber den Verlauf der zahlentheoretischer 
               Function sigma(n) = Sum_{x=1..n} mu(x) im Intervalle von 0 bis 150 
               000, Sitzungsbericht der Kaiserlichen Akademie der Wissenschaften 
               Wien, Mathematisch-Naturwissenschaftlichen Klasse, 2a, v. 106, 1897, 
               835-1024.
%H A002321 T. D. Noe, <a href="b002321.txt">Table of n, a(n) for n = 1..10000</a>
%H A002321 G. J. Chaitin, <a href="http://arXiv.org/abs/math.HO/0306042">[math/0306042] 
               Thoughts on the Riemann hypothesis</a>
%H A002321 J. B. Conrey, <a href="http://www.ams.org/notices/200303/fea-conrey-web.pdf">
               The Riemann Hypothesis</a>
%H A002321 F. Dress, <a href="http://www.expmath.org/expmath/volumes/2/2.html">Fonction 
               sommatoire de la fonction de Moebius. 1. Majorations experimentales</
               a>.
%H A002321 F. Dress, <a href="http://www.expmath.org/expmath/volumes/2/2.html">Fonction 
               sommatoire de la fonction de Moebius. 2. Majorations asymptotiques 
               elementaires</a>.
%H A002321 M. El-Marraki, <a href="http://www.emis.de/journals/JTNB/1995-2/jtnb7-2_english.html#jourelec">
               Fonction sommatoire de la fonction mu de Moebius</a>
%H A002321 A. M. Odlyzko and H. J. J. te Riele, <a href="http://www.dtc.umn.edu/
               ~odlyzko/doc/zeta.html">Disproof of the Mertens conjecture</a>, J. 
               reine angew. Math., 357 (1985), pp. 138-160.
%H A002321 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/
               TABLES/aaaFArit/MobiusMe.htm">Nombres de Moebius et de Mertens</a>
%H A002321 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MertensFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A002321 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RedhefferMatrix.html">Redheffer Matrix</a>
%H A002321 Wikipedia, <a href="http://en.wikipedia.org/wiki/Mertens_function">Mertens 
               function</a>
%F A002321 Assuming the Riemann hypothesis, a(n) = O(x^(1/2 + eps)) for every eps 
               > 0 (Littlewood - see Landau p. 161).
%p A002321 with(numtheory); A002321 := n->add(mobius(k),k=1..n);
%t A002321 Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ]
%o A002321 (PARI) a(n)=sum(k=1,n,moebius(k))
%o A002321 (PARI) a(n)=if(n<1,0,matdet(matrix(n,n,i,j,(j==1)|(0==j%i))))
%Y A002321 Cf. A008683, A059571.
%Y A002321 A152901 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]
%Y A002321 Sequence in context: A160520 A145866 A103318 this_sequence A043530 A164995 
               A055718
%Y A002321 Adjacent sequences: A002318 A002319 A002320 this_sequence A002322 A002323 
               A002324
%K A002321 sign,easy,nice
%O A002321 1,5
%A A002321 N. J. A. Sloane (njas(AT)research.att.com).

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