1,1

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 (jvospost3(AT)gmail.com), Jun 11 2008

Denote by lambda(n) Liouville's function concerning the parity of the number of prime divisors of n. Using a theorem of Allouche, Mendes France and Peyriere and many classical results from the theory of the distribution of prime numbers, we prove that lambda(n) is not k-automatic for any k > 2. This yields that sum[n=1..infty] lambda(n) X^n an element of F_p[[X]] is transcendental over F_p(X) for any prime p > 2. Similar results are proved (or reproved) for many common number--theoretic functions, including phi, mu, Omega, omega, rho and others. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 22 2008]

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

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.

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

P. Ribenboim, Algebraic Numbers, p. 44.

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

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

Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions

Michael Coons, (Non)Automaticity of number theoretic functions, Oct 21, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 22 2008]

Weisstein, Eric W., Liouville Function. [From Daniel Forgues (squid(AT)zensearch.com), Mar 17 2009]

Dirichlet g.f.: zeta(2s)/zeta(s).

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 (jvospost3(AT)gmail.com), Apr 16 2006

a(n) = A033999(A001222(n). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 28 2009]

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:

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

Cf. A002053, A007421, A002819, A026424, A028260, A028488, A056912, A056913, A001222, A065043, A066829.

Cf. A001222.

Sequence in context: A164660 A114523 A000012 this_sequence A064179 A106400 A112865

Adjacent sequences: A008833 A008834 A008835 this_sequence A008837 A008838 A008839

sign,easy,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).