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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.