%I #133 Jun 22 2024 16:17:23

%S 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,

%T -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,

%U 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

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

%C 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

%C Coons proves that a(n) is not k-automatic for any k > 2. - _Jonathan Vos Post_, Oct 22 2008

%C 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

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

%D 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.

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

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

%D P. Ribenboim, Algebraic Numbers, p. 44.

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

%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 112.

%H T. D. Noe, <a href="/A008836/b008836.txt">Table of n, a(n) for n = 1..10000</a>

%H P. Borwein, R. Ferguson, and M. J. Mossinghoff, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02036-X">Sign changes in sums of the Liouville function</a>, Math. Comp. 77 (2008), 1681-1694.

%H Benoit Cloitre, <a href="http://arxiv.org/abs/1107.0812">A tauberian approach to RH</a>, arXiv:1107.0812 [math.NT], 2011.

%H Michael Coons and Peter Borwein, <a href="http://arxiv.org/abs/0806.1563">Transcendence of Power Series for Some Number Theoretic Functions</a>, arXiv:0806.1563 [math.NT], 2008.

%H Michael Coons, <a href="http://arxiv.org/abs/0810.3709">(Non)Automaticity of number theoretic functions</a>, arXiv:0810.3709 [math.NT], 2008.

%H H. Gupta, <a href="/A002053/a002053.pdf">On a table of values of L(n)</a>, Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]

%H R. S. Lehman, <a href="http://dx.doi.org/10.1090/S0025-5718-1960-0120198-5">On Liouville's function</a>, Math. Comp., 14 (1960), 311-320.

%H Andrei Vieru, <a href="http://arxiv.org/abs/1306.0496">Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions</a>, arXiv:1306.0496 [math.GM], 2015.

%H H. Walum, <a href="http://dx.doi.org/10.1016/0022-314X(80)90072-4">A recurrent pattern in the list of quadratic residues mod a prime and in the values of the Liouville lambda function</a>, J. Numb. Theory 12 (1) (1980) 53-56.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Liouville_function">Liouville function</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

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

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

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

%F a(n) = (-1)^A001222(n) = (-1)^bigomega(n). - _Jonathan Vos Post_, Apr 16 2006

%F a(n) = A033999(A001222(n)). - _Jaroslav Krizek_, Sep 28 2009

%F Sum_{d|n} a(d) *(A000005(d))^2 = a(n) *Sum{d|n} A000005(d). - _Vladimir Shevelev_, May 22 2010

%F a(n) = 1 - 2*A066829(n). - _Reinhard Zumkeller_, Nov 19 2011

%F a(n) = i^(tau(n^2)-1) where tau(n) is A000005 and i is the imaginary unit. - _Anthony Browne_, May 11 2016

%F a(n) = A106400(A156552(n)). - _Antti Karttunen_, May 30 2017

%F Recurrence: a(1)=1, n > 1: a(n) = sign(1/2 - Sum_{d<n, d|n} a(d)). - _Mats Granvik_, Oct 11 2017

%F a(n) = Sum_{ d | n } A008683(d)*A010052(n/d). - _Jinyuan Wang_, Apr 20 2019

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

%F a(n) = (-1)^A349905(n). - _Antti Karttunen_, Apr 26 2022

%F From _Ridouane Oudra_, Jun 02 2024: (Start)

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

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

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

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

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

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

%p A008836 := n -> (-1)^numtheory[bigomega](n); # _Peter Luschny_, Sep 15 2011

%p 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:

%t Table[LiouvilleLambda[n], {n, 100}] (* _Enrique Pérez Herrero_, Dec 28 2009 *)

%t Table[If[OddQ[PrimeOmega[n]],-1,1],{n,110}] (* _Harvey P. Dale_, Sep 10 2014 *)

%o (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 */

%o (PARI) a(n)=(-1)^bigomega(n) \\ _Charles R Greathouse IV_, Jan 09 2013

%o (Haskell)

%o a008836 = (1 -) . (* 2) . a066829 -- _Reinhard Zumkeller_, Nov 19 2011

%o (Python)

%o from sympy import factorint

%o def A008836(n): return -1 if sum(factorint(n).values()) % 2 else 1 # _Chai Wah Wu_, May 24 2022

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

%Y Möbius transform of A010052.

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

%K sign,easy,nice,mult

%O 1,1

%A _N. J. A. Sloane_