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A008836
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Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).
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42
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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; internal format)
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OFFSET
| 1,1
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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 (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]
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REFERENCES
| T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
Peter Borwein, Ron Ferguson and Michael J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694 [From Enrique Perez Herrero (eperezherrero(AT)yahoo.es), Dec 28 2009]
B. Cloitre, A tauberian approach to RH, Arxiv preprint arXiv:1107.0812, 2011
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.
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LINKS
| 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, arXiv:0806.1563
Michael Coons, (Non)Automaticity of number theoretic functions, Oct 21, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 22 2008]
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Weisstein, Eric W., Liouville Function. [From Daniel Forgues (squid(AT)zensearch.com), Mar 17 2009]
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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 (jvospost3(AT)gmail.com), Apr 16 2006
a(n) = A033999(A001222(n). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 28 2009]
Sum{d|n} a(d) *(A000005(d))^2 = a(n) *Sum{d|n} A000005(d). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 22 2010]
a(n) = 1 - 2*A066829(n). [Reinhard Zumkeller, Nov 19]
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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:
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MATHEMATICA
| A008836[n_]:=LiouvilleLambda[n] [From Enrique Perez Herrero (eperezherrero(AT)yahoo.es), Dec 28 2009]
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PROG
| (PARI) a(n)=if(n<1, 0, n=factor(n); (-1)^sum(i=1, matsize(n)[1], n[i, 2]))
(Haskell)
a008836 = (1 -) . (* 2) . a066829 -- Reinhard Zumkeller, Nov 19
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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
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KEYWORD
| sign,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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