A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice
• mult

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..infinity} f(n)z^n is transcendental over {Z}[z]; in particular, sum_{n=1..infinity} lambda(n)z^n is transcendental, where lambda is Liouville's function. The transcendence of sum_{n=1..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

Examples

			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.
		

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.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.3.5 on page 99.
• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 112.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• P. Borwein, R. Ferguson, and M. J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.
• Benoit 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], 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.
• H. Walum, A recurrent pattern in the list of quadratic residues mod a prime and in the values of the Liouville lambda function, J. Numb. Theory 12 (1) (1980) 53-56.
• Eric Weisstein's World of Mathematics, Liouville Function
• Wikipedia, Liouville function
• Index to divisibility sequences
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000005, A001222, A002053, A007421, A002819 (partial sums), A008683, A010052, A026424, A028260, A028488, A056912, A056913, A065043, A066829, A106400, A156552, A349905, A063647, A101455, A048691.

Möbius transform of A010052.

Cf. A182448 (Dgf at s=2), A347328 (Dgf at s=3), A347329 (Dgf at s=4).

Programs

Haskell

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

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 *)

PARI

{a(n) = if( n<1, 0, n=factor(n); (-1)^sum(i=1, matsize(n)[1], n[i,2]))}; /* Michael Somos, Jan 01 2006 */

PARI

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

Python

from sympy import factorint
def A008836(n): return -1 if sum(factorint(n).values()) % 2 else 1 # Chai Wah Wu, May 24 2022

Formula

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

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

a(n) = A106400(A156552(n)). - Antti Karttunen, May 30 2017

Recurrence: a(1)=1, n > 1: a(n) = sign(1/2 - Sum_{dMats Granvik, Oct 11 2017

a(n) = Sum_{ d | n } A008683(d)*A010052(n/d). - Jinyuan Wang, Apr 20 2019

a(1) = 1; a(n) = -Sum_{d|n, d < n} mu(n/d)^2 * a(d). - Ilya Gutkovskiy, Mar 10 2021

a(n) = (-1)^A349905(n). - Antti Karttunen, Apr 26 2022

From Ridouane Oudra, Jun 02 2024: (Start)

a(n) = (-1)^A066829(n);

a(n) = (-1)^A063647(n);

a(n) = A101455(A048691(n));

a(n) = sin(tau(n^2)*Pi/2). (End)