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See a nice lecture by Terence Tao on some deep questions concerning multiplicative functions and more.
Are there any other functions that would work? Edit: I guess you could just flip the sign of a large prime.
Here it's said, that function
is asymptotically ok if p is odd prime (i mean x is coprime with p, and modulo is 1 or -1, not 1 or p - 1). The function in post, is this function for p = 3. I'm not sure constant is good enough for other primes.