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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
54
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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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Also determinant of n X n (0,1) matrix defined by A(i,j)=1 if j=1 or i divides j.
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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.
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.
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.
Sequence in context: A092505 A066086 A103318 this_sequence A043530 A055718 A007302
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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njas
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