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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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101
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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; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| 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, 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, Jul 19 2009]
Determinant of n X n (-1,0,1) matrix defined by: if j=1 then A(i,j)=1 elseif i divides j then A(i,j)=0 else A(i,j)=-1. [From Mats Granvik, Apr 23 2010]
The first positive value of Mertens function for n > 1 is for n = 94. The graph seems to show a negative bias for the Mertens function which is eerily similar to the Chebyshev bias (described in A156749 and A156709) The purported bias seems to be empirically approximated (by looking at the graph)
- (6 / pi^2) * (sqrt(n) / 4)
where 6 / pi^2 = 1 / zeta(2) is the asymptotic density of squarefree numbers (the squareful numbers having Moebius mu of 0). This would be a growth pattern akin to the Chebyshev bias. - Daniel Forgues, Jan 23 2011
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REFERENCES
| 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).
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in THEORETICAL ASPECTS OF COMPUTING, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15; http://www.springerlink.com/content/x0044261g7004717/.
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.
J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge, 1999, see p. 482.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
B. Boncompagni, Selected values of the Mertens function
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, Mertens Function
Eric Weisstein's World of Mathematics, Redheffer Matrix
Wikipedia, Mertens function
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FORMULA
| Assuming the Riemann hypothesis, a(n) = O(x^(1/2 + eps)) for every eps > 0 (Littlewood - see Landau p. 161).
Lambert series: Sum_{n >= 1} A002321(n)*(x^n/(1-x^n)-x^(n+1)/(1-x^(n+1))) =x and -1/x. [From Mats Granvik, Sep 09 2010]
a(n)+2 = A192763(n,1) for n>1, and A192763(1,k) for k>1 (conjecture). [From Mats Granvik, Jul 10 2011]
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EXAMPLE
| x - x^3 - x^4 - 2*x^5 - x^6 - 2*x^7 - 2*x^8 - 2*x^9 - x^10 - 2*x^11 - 2*x^12 - ...
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MAPLE
| with(numtheory); A002321 := n->add(mobius(k), k=1..n);
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MATHEMATICA
| Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ]
Accumulate[Array[MoebiusMu, 100]] (* From Harvey P. Dale, May 11 2011 *)
(*Conjectured recurrence (two combined recurrences):*)
t[n_, k_] := t[n, k] = If[And[n == 1, k == 1], 3, If[Or[And[n == 1, k == 2], And[n == 2, k == 1]], 2, If[n == 1, (-t[n, k - 1] - Sum[t[i, k], {i, 2, k - 1}])/(k + 1) + t[n, k - 1], If[k == 1, (-t[n - 1, k] - Sum[t[n, i], {i, 2, n - 1}])/(n + 1) + t[n - 1, k], If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[t[k - i, n], {i, 1, n - 1}]]]]]];
nn = 81;
MatrixForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]];
Table[t[1, k], {k, 1, nn}] - 2 (* From Mats Granvik, Jul 10, 2011 *)
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PROG
| (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
| Cf. A008683, A059571.
A152901 [From Gary W. Adamson, Dec 14 2008]
First column of A134541. [From Mats Granvik, Apr 24 2010]
First column of A179287. [From Mats Granvik, Jul 09 2010]
Sequence in context: A160520 A145866 A103318 * A043530 A164995 A055718
Adjacent sequences: A002318 A002319 A002320 * A002322 A002323 A002324
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KEYWORD
| sign,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Cross reference (Aug 28 2010) deleted by Mats Granvik (mats.granvik(AT)abo.fi), Sep 11 2010
-1/x added to Lambert series by Mats Granvik (mats.granvik(AT)abo.fi), Sep 23 2010
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