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 A002321 Mertens's function: Sum_{k=1..n} mu(k), where mu is the Moebius function A008683. (Formerly M0102 N0038) 113
 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; text; internal format)
 OFFSET 1,5 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. The first positive value of Mertens's 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 to - (6 / Pi^2) * (sqrt(n) / 4) (by looking at the graph) (see MathOverflow link, May 28 2012) 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 All integers appear infinitely often in this sequence. - Charles R Greathouse IV, Aug 06 2012 Soundararajan proves that, on the Riemann Hypothesis, a(n) << sqrt(n) exp(sqrt(log n)*(log log n)^14), sharpening the well-known equivalence. - Charles R Greathouse IV, Jul 17 2015 REFERENCES E. Landau, Vorlesungen über 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. F. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897), p. 761-830. D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.1. N. C. Ng, The summatory function of the Mobius function, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-316. 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. J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge, 1999, see p. 482. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 B. Boncompagni, Selected values of the Mertens function O. Bordelles, Some Explicit Estimates for the Mobius Function , J. Int. Seq. 18 (2015) 15.11.1 G. J. Chaitin, Thoughts on the Riemann hypothesis, arXiv:math/0306042 [math.HO],  2003. J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347. Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295. F. Dress, Fonction sommatoire de la fonction de Moebius. 1. Majorations expérimentales, Experiment. Math. , Volume 2, Issue 2 (1993), 89-98. F. Dress, M. El Marraki, Fonction sommatoire de la fonction de Moebius. 2. Majorations asymptotiques élémentaires, Experiment. Math., Volume 2, Issue 2 (1993), 99-112. M. El-Marraki, Fonction sommatoire de la fonction mu de Möbius, 3. Majorations asymptotiques effectives fortes, Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 2 , p. 407-433. MathOverflow, Is Mertens function negatively biased?, posted May 28, 2012 MathOverflow, Approximations to the Mertens function, posted Jul 08 2015 Nathan Ng, The distribution of the summatory function of the Möbius function, arXiv:math/0310381 [math.NT], 2003. A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. reine angew. Math., 357 (1985), pp. 138-160. Lowell Schoenfeld, An improved estimate for the summatory function of the Möbius function, Acta Arithmetica 15:3 (1969), pp. 221-233. Kannan Soundararajan, Partial sums of the Möbius function, arXiv:0705.0723 [math.NT], 2007-2008. Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, Jun 05 2014, Pages 105-124. 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 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 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} a(n)*(x^n/(1-x^n)-x^(n+1)/(1-x^(n+1))) = x and -1/x. - Mats Granvik, Sep 09 2010 a(n)+2 = A192763(n,1) for n>1, and A192763(1,k) for k>1 (conjecture). - Mats Granvik, Jul 10 2011 Sum_{k = 1..n} a(floor(n/k)) = 1. - David W. Wilson, Feb 27 2012 a(n) = Sum_{k = 1..n} tau_{-2}(k) * floor(n/k), where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 23 2013 a(n) = Sum_{k=1..A002088(n)} exp(2*Pi*i*A038566(k)/A038567(k-1)) where i is the imaginary unit. - Eric Desbiaux, Jul 31 2014 Schoenfeld proves that |a(n)| < 5.3*n/(log n)^(10/9) for n > 1. - Charles R Greathouse IV, Jan 17 2018 EXAMPLE G.f. = 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 - ... MAPLE with(numtheory); A002321 := n->add(mobius(k), k=1..n); MATHEMATICA Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ] Accumulate[Array[MoebiusMu, 100]] (* 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 (* Mats Granvik, Jul 10, 2011 *) 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))) (PARI) a(n)=my(s); forsquarefree(k=1, n, s+=moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018 (Haskell) import Data.List (genericIndex) a002321 n = genericIndex a002321_list (n-1) a002321_list = scanl1 (+) a008683_list -- Reinhard Zumkeller, Jul 14 2014, Dec 26 2012 (Python) from sympy import mobius def M(n): return sum([mobius(k) for k in xrange(1, n + 1)]) print [M(n) for n in xrange(1, 151)] # Indranil Ghosh, Mar 18 2017 CROSSREFS Cf. A008683, A059571, A209802. First column of A134541. First column of A179287. Sequence in context: A297770 A145866 A103318 * A043530 A297771 A164995 Adjacent sequences:  A002318 A002319 A002320 * A002322 A002323 A002324 KEYWORD sign,easy,nice AUTHOR EXTENSIONS Cross reference (Aug 28 2010) deleted by Mats Granvik, Sep 11 2010 -1/x added to Lambert series by Mats Granvik, Sep 23 2010 STATUS approved

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Last modified November 16 02:33 EST 2018. Contains 317252 sequences. (Running on oeis4.)