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 nontrivial 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
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
Peter Borwein, Ron Ferguson and Michael 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), Proc. Indian Acad. Sci. Sect. A, 12 (1940), 407-409. [Annotated scanned copy]
R. Sherman Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Maxie Dion Schmidt, Picking up the partial sums of the Möbius function problem with probabilistic number theory, arXiv:2604.23517 [math.NT], 2026. See p. 1 (Def. 1.2).
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.
Herbert 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.
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) = 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
Recurrence: a(1)=1, n > 1: a(n) = sign(1/2 - Sum_{d<n, d|n} a(d)). - Mats Granvik, Oct 11 2017
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) = sin(tau(n^2)*Pi/2). (End)
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
# Alternative:
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]))}; /* Michael Somos, Jan 01 2006 */
(PARI) a(n)=(-1)^bigomega(n) \\ Charles R Greathouse IV, Jan 09 2013
(Haskell)
a008836 = (1 -) . (* 2) . a066829 -- Reinhard Zumkeller, Nov 19 2011
(Python)
from sympy import factorint
def A008836(n): return -1 if sum(factorint(n).values()) % 2 else 1 # Chai Wah Wu, May 24 2022
CROSSREFS
KEYWORD
sign,easy,nice,mult
AUTHOR
STATUS
approved
