I wonder if there is any efficient way to calculate Möbius function for a array of number 1:1000000 


<http://en.wikipedia.org/wiki/M%C3%B6bius_function>



calculating Möbius function
Here are a few papers that might be helpful. 


Shallit and Shamir, Number-theoretic functions which are equivalent to number of divisors, Inform. Process. Lett. 20 (1985), no. 3, 151–153, MR0801982 (86k:11076). The review, by Hale Trotter, says $\mu(n)$ can be calculated from a single value $d(n^q)$ where $d$ is the divisor function and $q$ is a prime greater than $1+\log\_2n$. 


Lioen and van de Lune, Systematic computations on Mertens' conjecture and Dirichlet's divisor problem by vectorized sieving, in From Universal Morphisms to Megabytes: a Baayen Space Odyssey, 421–432, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1994, MR1490603 (98j:11125). The review, by Jonathan P. Sorenson, says the authors present sieving algorithms for computing $\mu(n)$. 


Herman te Riele, Computational sieving applied to some classical number-theoretic problems, in Number Theory in Progress, Vol. 2 (Zakopane-Kościelisko, 1997), 1071–1080, de Gruyter, Berlin, 1999, MR1689561 (2000e:11119). The review, by Marc Deléglise, starts, "This paper is a survey about different applications of sieving in number theory. Finding all the primes belonging to a given interval and computing all the values of the Möbius function on a given interval are obvious examples." 


