A084237 -id:A084237 - cp's OEIS frontend

A002321 Mertens's function: Sum_{k=1..n} mu(k), where mu is the Moebius function A008683.

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

Offset: 1

N. J. A. Sloane

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
Balazard & De Roton improve this (on the Riemann Hypothesis) to a(n) << sqrt(n) exp(sqrt(log n)*(log log n)^k) for any k > 5/2, where the implied constant in the Vinogradov symbol depends on k. Saha & Sankaranarayanan reduce the exponent to 5/4 on additional hypotheses. - Charles R Greathouse IV, Feb 02 2023

			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 - ...

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.
Biswajyoti Saha and Ayyadurai Sankaranarayanan, On estimates of the Mertens function, International Journal of Number Theory, Vol. 15, No. 02 (2019), pp. 327-337.
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).
J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge, 1999, see p. 482.

T. D. Noe, Table of n, a(n) for n = 1..10000
Michel Balazard and Anne De Roton, Sur un critère de Baez-Duarte pour l'hypothèse de Riemann, International Journal of Number TheoryVol. 06, No. 04, pp. 883-903 (2010). arXiv preprint (2008). arXiv:0812.1689 [math.NT]
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 and 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.
Brady Haran, Holly Krieger, and Pete McPartlan, A Prime Surprise (Mertens Conjecture), Numberphile video (2019).
Harald A. Helfgott and Lola Thompson, Summing mu(n): a faster elementary algorithm, arXiv:2101.08773 [math.NT], 2021.
Greg Hurst, Computations of the Mertens function and improved bounds on the Mertens conjecture, arXiv:1610.08551 [math.NT], 2016-2017.
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, Proc. London Math. Soc. (3) 89 (2004), no. 2, 361-389; 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, Journal für die reine und angewandte Mathematik, Vol. 631 (2009), pp. 141-152. arXiv:0705.0723 [math.NT], 2007-2008.
Robert Daublebsky 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.
Robert Daublebsky von Sterneck, Bemerkung über die Summierung einiger zahlen-theoretischen Functionen, Monatshefte für Mathematik und Physik 9(1) (1898), 43-45. [He proves the inequality |a(n)| <= (n/9) + 8.]
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.
H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6.

Cf. A008683, A059571, A084237, A209802.
First column of A134541.
First column of A179287.

import Data.List (genericIndex)
a002321 n = genericIndex a002321_list (n-1)
a002321_list = scanl1 (+) a008683_list
-- Reinhard Zumkeller, Jul 14 2014, Dec 26 2012

[&+[MoebiusMu(k): k in [1..n]]: n in [1..81]]; // Bruno Berselli, Jul 12 2021

with(numtheory); A002321 := n->add(mobius(k),k=1..n);

Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ]
Accumulate[Array[MoebiusMu,100]] (* Harvey P. Dale, May 11 2011 *)

PARI
```
a(n) = sum( k=1, n, moebius(k))
```

a(n) = if( n<1, 0, matdet( matrix(n, n, i, j, j==1 || 0==j%i)))

a(n)=my(s); forsquarefree(k=1,n, s+=moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy import mobius
def M(n): return sum(mobius(k) for k in range(1,n + 1))
print([M(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 18 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A002321(n):
    if n == 0:
        return 0
    c, j = n, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A002321(k1)
        j, k1 = j2, n//j2
    return j-c # Chai Wah Wu, Mar 30 2021

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 and Sep 23 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
G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x - Sum_{k>=2} (1 - x^k) * A(x^k)). - Ilya Gutkovskiy, Aug 11 2021

A087987 a(n) is the Mertens function value at the n-th primorial number.

Original entry on oeis.org

0, -1, -3, -1, -1, 16, -25, 278, 3516, -5012, -30431, -234676, -4247453, -6271957, 177533446, -416214476

Offset: 1

Labos Elemer, Oct 02 2003

			n = 3: 3rd primorial number = 30. A002321(30) = -3.

Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.

Cf. A002110, A002321, A008683, A084237.

q[x_] := Apply[Times, Table[Prime[i], {i, 1, x}]] s=0; i=1; Do[While[i<=q[n], s=s+MoebiusMu[i]; i++ ]; Print[s], {n, 0, 8}]//Timing

p=1; s=1; for(n=1,10,pr=p; p*=prime(n); s+=sum(k=pr+1, p, moebius(k)); print1(s",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 21 2007

use ntheory ":all"; say mertens(pn_primorial($)) for 1..12; # _Dana Jacobsen, May 22 2015

a(n) = A002321(A002110(n)).

One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 21 2007
a(10)-a(11) from Donovan Johnson, Jun 21 2012
a(12)-a(13) from Rikard Nordgren, Nov 11 2012
a(14) from Chai Wah Wu, Apr 24 2021
a(15)-a(16) from Henri Lifchitz, Nov 10 2024

A348708 Number of squarefree integers with an odd number of prime factors <= 10^n.

Original entry on oeis.org

0, 4, 30, 303, 3053, 30421, 303857, 3039127, 30395383, 303963673, 3039652332, 30396399068, 303963519954, 3039635209356, 30396355530761, 303963552535238, 3039635510867921, 30396355103617024, 303963550950390745, 3039635508820145344, 30396355092470750098, 303963550928711270447

Offset: 0

Giorgos Kalogeropoulos, Oct 30 2021

nonn

a(n) is the number of integers k <= 10^n with mu(k)=-1 where mu(k) is the Möbius function.

			a(1) = 4 because there are 4 squarefree integers with an odd number of prime factors <= 10: 2, 3, 5, 7.

Cf. A071172, A084237, A008683, A030059, A104141, A348709.

Table[Length@Select[Range[10^n],MoebiusMu@#==-1&],{n,0,6}]

a(n) = (A071172(n) - A084237(n)) / 2.

Lim_{n->oo} a(n)/10^n = 3/Pi^2 (A104141).

A348709 Number of squarefree integers with an even number of prime factors <= 10^n.

Original entry on oeis.org

1, 3, 31, 305, 3030, 30373, 304069, 3040164, 30397311, 303963451, 3039618610, 30396311212, 303963582320, 3039635808938, 30396354655186, 303963549318865, 3039635507672484, 30396355081786770, 303963550903632005, 3039635509720135531, 30396355092931863204, 303963550925315375170

Offset: 0

Giorgos Kalogeropoulos, Oct 30 2021

nonn

a(n) is the number of integers k <= 10^n with mu(k)=1 where mu(k) is the Möbius function.

			a(1) = 3 because there are 3 squarefree integers with an even number of prime factors <= 10: 1, 6, 10.

Cf. A071172, A084237, A008683, A030229, A104141, A348708.

Table[Length@Select[Range[10^n],MoebiusMu@#==1&],{n,0,6}]

a(n) = (A071172(n) + A084237(n)) / 2.

Lim_{n->oo} a(n)/10^n = 3/Pi^2 (A104141).

A087989 a(n) = M(n!), the value of Mertens's function at the n-th factorial.

Original entry on oeis.org

1, 1, 0, -1, -2, -3, -3, -6, -15, -138, -26, -527, -3474, -19550, -20014, 3084, -1253696, 7121000, -18636176, -44667415, 94933922, -848322099

Offset: 0

Labos Elemer, Oct 02 2003

			a(4) = A002321(4!) = A002321(24) = -2.

Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.

Cf. A000142, A002321, A084237, A087987.

s=0; i=1; Do[While[i<= n!, s=s+MoebiusMu[i]; i++ ]; Print[s], {n, 0, 10}]

a(n)=sum(k=1,n!,moebius(k)) \\ Charles R Greathouse IV, Apr 02 2014

a(n) = A002321(A000142(n)).

More terms from Sean A. Irvine, Dec 06 2009
a(14) from Donovan Johnson, Jun 21 2012
a(15)-a(17) from Rikard Nordgren, Nov 10 2012
a(18)-a(21) from Henri Lifchitz, Nov 10 2024

A189020 a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).

Original entry on oeis.org

1, 89, 3575, 93237, 1951526, 35270969, 578262093, 8840109380, 128217432396, 1784942188189, 24045237260214, 315312623543840, 4042957241191810, 50862246063060180, 629513636928477232, 7681900592647818929, 92587253467765253144, 1103781870246459696784, 13031388731053572679450, 152516435040764735691556, 1771079109308495896176156

Offset: 0

Andrew Lelechenko, Apr 15 2011

nonn

Using that tau_4 = tau_2 ** tau_2, where ** means Dirichlet convolution and tau_2 is (A000005), one can calculate a(n) faster than in O(10^n) operations - namely in O(10^(3n/4)) or even in O(10^(2n/3)). See links for details.

A. V. Lelechenko, The summation of the multiplicative functions (in Russian).

Cf. A057494 - partial sums up to 10^n of the divisors function tau_2 (A000005), A180361 - of the unitary divisors function tau_2* (A034444), A180365 - of the 3-divisors function tau_3 (A007425).

Also see A072692 for such sums of the sum of divisors function (A000203), A084237 for sums of Moebius function (A008683), A064018 for sums of Euler totient function (A000010).

a(n) = A061202(10^n) = Sum_{k=1..10^n} A007426(n).

a(16)-a(20) from Henri Lifchitz, Feb 05 2025

A201740 The value of the Mertens function at n^n.

Original entry on oeis.org

1, -1, -1, -1, 4, 39, -158, 211, -186, -33722, 55130, 192039, -4032991

Offset: 1

Carl Najafi, Dec 04 2011

Marc Deleglise and Joel Rivat, Computing the summation of the Mobius function, Experiment. Math. 5 (1996), no. 4, 291-295.

Cf. A002321, A000312, A084237.

Table[Sum[MoebiusMu[k], {k, n^n}], {n, 7}]

a(n) = A002321(n^n).

a(9)-a(10) from Max Alekseyev, Dec 10 2011
a(11) from Donovan Johnson, Jun 21 2012
a(12)-a(13) from Rikard Nordgren, Nov 10 2012

cp's OEIS Frontend

A002321 Mertens's function: Sum_{k=1..n} mu(k), where mu is the Moebius function A008683.

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A087987 a(n) is the Mertens function value at the n-th primorial number.

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A348708 Number of squarefree integers with an odd number of prime factors <= 10^n.

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A348709 Number of squarefree integers with an even number of prime factors <= 10^n.

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A087989 a(n) = M(n!), the value of Mertens's function at the n-th factorial.

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A189020 a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).

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A201740 The value of the Mertens function at n^n.

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