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A002321 Mertens' function: Sum_{1<=k<=n} mu(k), where mu is the Moebius function A008683.
(Formerly M0102 N0038)
104
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.

Equals row sums of triangle A152901. [Gary W. Adamson, Dec 14 2008]

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 to - (6 / Pi^2) * (sqrt(n) / 4) (by looking at the graph) (Cf. link to MathOverflow post on 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

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

G. J. Chaitin, [math/0306042] Thoughts on the Riemann hypothesis

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 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

MathOverflow, Is Mertens function negatively biased?, posted May 28, 2012

A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. reine angew. Math., 357 (1985), pp. 138-160.

Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 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([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(exp(2*Pi*i*A038566(k)/A038567(k-1)), k=1..A002088(n), where i is the imaginary unit. - Eric Desbiaux, Jul 31 2014

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))))};

(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

CROSSREFS

Cf. A008683, A059571, A152901, A209802.

First column of A134541.

First column of A179287.

Sequence in context: A235708 A145866 A103318 * A043530 A164995 A055718

Adjacent sequences:  A002318 A002319 A002320 * A002322 A002323 A002324

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

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 December 19 18:23 EST 2014. Contains 252239 sequences.