Question

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

Answer 1

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."

Answer 2

For range 1:1000000 the stupidest possible algorithm (factor each integer, check if it's square free, raise -1 to the right power) will terminate in a blink of an eye. For much larger ranges, an obvious modifican of the sieve of erathsosphenes will be very fast ( just remember to zero every $p$th number at the $p$th step.

Answer 3

You don't need the bigger machinery of a segmented sieve for such a small range. Here is a simple $O(n\log\log n)$ algorithm to calculate all $\mu(i)$ up to $n$ based on the Sieve of Eratosthenes. In fact, the sieve does fully factor all the square-free numbers; it just doesn't do it one at a time.

Depending on your computer, this approach is practical up to around $2^{30}$ at which point you need to start using higher-precision arithmetic and computing a range of $\mu(i)$ values in batches.

public static int[] GetMu(int max)
{
    var sqrt = (int)Math.Floor(Math.Sqrt(max));
    var mu = new int[max + 1];
    for (int i = 1; i <= max; i++)
        mu[i] = 1;
    for (int i = 2; i <= sqrt; i++)
    {
        if (mu[i] == 1)
        {
            for (int j = i; j <= max; j += i)
                mu[j] *= -i;
            for (int j = i * i; j <= max; j += i * i)
                mu[j] = 0;
        }
    }
    for (int i = 2; i <= max; i++)
    {
        if (mu[i] == i)
            mu[i] = 1;
        else if (mu[i] == -i)
            mu[i] = -1;
        else if (mu[i] < 0)
            mu[i] = 1;
        else if (mu[i] > 0)
            mu[i] = -1;
    }
    return mu;
}

Running GetMu(1000000) takes about 10 msec on my computer.