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Search: id:A002321
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| A002321 |
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Mertens's function: Sum_{1<=k<=n} mu(k), where mu = Moebius function (A008683).
(Formerly M0102 N0038)
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+0
70
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| 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, -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, -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
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Partial sums of the Moebius function A008683.
Also determinant of n X n (0,1) matrix defined by A(i,j)=1 if j=1 or i divides j.
Equals row sums of triangle A152901 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]
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]
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REFERENCES
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J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347.
Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius function. Experiment. Math. 5 (1996), no. 4, 291-295.
E. Landau, Vorlesungen ueber Zahlentheorie, Chelsea, NY, Vol. 2, p. 157.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
N. C. Ng, The summatory function of the Mobius function, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-316.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
G. J. Chaitin, [math/0306042] Thoughts on the Riemann hypothesis
J. B. Conrey, The Riemann Hypothesis
F. Dress, Fonction sommatoire de la fonction de Moebius. 1. Majorations experimentales.
F. Dress, Fonction sommatoire de la fonction de Moebius. 2. Majorations asymptotiques elementaires.
M. El-Marraki, Fonction sommatoire de la fonction mu de Moebius
A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. reine angew. Math., 357 (1985), pp. 138-160.
G. Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Redheffer Matrix
Wikipedia, Mertens function
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FORMULA
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Assuming the Riemann hypothesis, a(n) = O(x^(1/2 + eps)) for every eps > 0 (Littlewood - see Landau p. 161).
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MAPLE
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with(numtheory); A002321 := n->add(mobius(k), k=1..n);
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MATHEMATICA
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Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ]
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PROGRAM
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(PARI) a(n)=sum(k=1, n, moebius(k))
(PARI) a(n)=if(n<1, 0, matdet(matrix(n, n, i, j, (j==1)|(0==j%i))))
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CROSSREFS
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Cf. A008683, A059571.
A152901 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]
Sequence in context: A160520 A145866 A103318 this_sequence A043530 A164995 A055718
Adjacent sequences: A002318 A002319 A002320 this_sequence A002322 A002323 A002324
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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