A318608 Moebius function mu(n) defined for the Gaussian integers.

1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0

Offset: 1

Author

Jianing Song, Aug 30 2018

Keywords

• sign
• easy
• mult

Comments

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Gaussian prime factor, otherwise (-1)^t if n is a product of a Gaussian unit and t distinct Gaussian prime factors.

a(n) = 0 for even n since 2 = -i*(1 + i)^2 contains a squared factor. For rational primes p == 1 (mod 4), p is always factored as (x + y*i)(x - y*i), x + y*i and x - y*i are not associated so a(p) = (-1)*(-1) = 1.

Interestingly, a(n) and A091069(n) have the same absolute value (= |A087003(n)|), since the discriminants of the quadratic fields Q[i] and Q[sqrt(2)] are -4 and 8 respectively, resulting in Q[i] and Q[sqrt(2)] being two of the three quadratic fields with discriminant a power of 2 or negated (the other one being Q[sqrt(-2)] with discriminant -8).

Examples

			a(15) = -1 because 15 is factored as 3*(2 + i)*(2 - i) with three distinct Gaussian prime factors.
a(21) = (-1)*(-1) = 1 because 21 = 3*7 where 3 and 7 are congruent to 3 mod 4 (thus being Gaussian primes).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Wikipedia, Gaussian integer

Crossrefs

Absolute values are the same as those of A087003.

First row and column of A103226.

Cf. A101455.

Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), this sequence ("mu", A008683).

Equivalent in the ring of Eisenstein integers: A319448.

Cf. A091069 (Moebius function over Z[sqrt(2)]).

Programs

Mathematica

f[p_, e_] := If[p == 2 || e > 1, 0, Switch[Mod[p, 4], 1, 1, 3, -1]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2020 *)

PARI

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2||e>=2, r=0);
        if(Mod(p,4)==3&e==1, r*=-1);
    );
    return(r);
}

Formula

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - (p mod 4)) where the product is taken over the primes.

Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == 1 (mod 4) and -1 if p == 3 (mod 4).

a(n) = 0 if A078458(n) != A086275(n), otherwise (-1)^A086275(n).

a(n) = A103226(n,0) = A103226(0,n).

For squarefree n, a(n) = Kronecker symbol (-4, n) = A101455(n). Also for these n, a(n) = A091069(n) if n even or n == 1 (mod 8), otherwise -A091069(n).