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A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity). 80
1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Coons and Borwein: "We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non-trivial completely multiplicative function from N to {-1,1), the series sum_{n=1 to infinity) f(n)z^n is transcendental over {Z}[z]; in particular, sum_{n=1 to infinity) lambda(n)z^n is transcendental, where lambda is Liouville's function. The transcendence of sum_{n=1 to infinity) mu(n)z^n is also proved." - Jonathan Vos Post, Jun 11 2008

Coons proves that a(n) is not k-automatic for any k > 2. - Jonathan Vos Post, Oct 22 2008

The Riemann hypothesis is equivalent to the statement that for every fixed epsilon > 0, lim n -> infinity (a(1) + a(2) + ... + a(n))/n^(1/2 + epsilon) = 0 (Borwein et al., theorem 1.2). - Arkadiusz Wesolowski, Oct 08 2013

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.

P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 1-11.

H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.

H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.

P. Ribenboim, Algebraic Numbers, p. 44.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

P. Borwein, R. Ferguson, M. J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.

B. Cloitre, A tauberian approach to RH, arXiv:1107.0812 [math.NT], 2011.

Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.

Michael Coons, (Non)Automaticity of number theoretic functions, arXiv:0810.3709 [math.NT], Oct 21, 2008.

H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]

R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Andrei Vieru, Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions, arXiv:1306.0496 [math.GM], 2015.

Eric Weisstein's World of Mathematics, Liouville Function

Wikipedia, Liouville function

Index to divisibility sequences

FORMULA

Dirichlet g.f.: zeta(2s)/zeta(s); Dirichlet inverse of A008966.

Sum_{ d divides n } lambda(d) = 1 if n is a square, else 0.

Completely multiplicative with a(p) = -1, p prime.

a(n) = (-1)^A001222(n) = (-1)^bigomega(n). - Jonathan Vos Post, Apr 16 2006

a(n) = A033999(A001222(n). - Jaroslav Krizek, Sep 28 2009

Sum{d|n} a(d) *(A000005(d))^2 = a(n) *Sum{d|n} A000005(d). - Vladimir Shevelev, May 22 2010

a(n) = 1 - 2*A066829(n). - Reinhard Zumkeller, Nov 19 2011

a(n) = i^(tau(n^2)-1) where tau(n) is A000005 and i is the imaginary unit. - Anthony Browne, May 11 2016

EXAMPLE

a(4) = 1 because since bigomega(4) = 2 (the prime divisor 2 is counted twice), then (-1)^2 = 1.

a(5) = -1 because 5 is prime and therefore bigomega(5) = 1 and (-1)^1 = -1.

MAPLE

A008836 := n -> (-1)^numtheory[bigomega](n); # Peter Luschny, Sep 15 2011

with(numtheory): A008836 := proc(n) local i, it, s; it := ifactors(n): s := (-1)^add(it[2][i][2], i=1..nops(it[2])): RETURN(s) end:

MATHEMATICA

Table[LiouvilleLambda[n], {n, 100}] (* Enrique Pérez Herrero, Dec 28 2009 *)

Table[If[OddQ[PrimeOmega[n]], -1, 1], {n, 110}] (* Harvey P. Dale, Sep 10 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, n=factor(n); (-1)^sum(i=1, matsize(n)[1], n[i, 2]))};

(PARI) a(n)=(-1)^bigomega(n) \\ Charles R Greathouse IV, Jan 09 2013

(Haskell)

a008836 = (1 -) . (* 2) . a066829  -- Reinhard Zumkeller, Nov 19 2011

CROSSREFS

Cf. A002053, A007421, A002819 (partial sums), A026424, A028260, A028488, A056912, A056913, A001222, A065043, A066829.

Sequence in context: A158387 * A087960 A164660 A106400 A112865 A114523

Adjacent sequences:  A008833 A008834 A008835 * A008837 A008838 A008839

KEYWORD

sign,easy,nice,mult,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 9 14:26 EST 2016. Contains 278971 sequences.