A319448 Moebius function mu(n) defined for the Eisenstein integers.

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

Offset: 1

Author

Jianing Song, Sep 19 2018

Keywords

• sign
• mult

Comments

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

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2. a(n) = 0 for n divisible by 3 since 3 = w'*(1 + w)^2 contains a squared factor. For rational primes p == 1 (mod 3), p is always factored as (x + y*w)(x + y*w'), x + y*w and x + y*w' are not associated so a(p) = (-1)*(-1) = 1.

Examples

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
a(14) = -1 because 14 is factored as 2*(2 + w)*(2 + w') with three distinct Eisenstein prime factors.
a(55) = (-1)*(-1) = 1 because 55 = 5*11 where 5 and 11 are congruent to 2 mod 3 (thus being Eisenstein primes).
		

Links

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

Crossrefs

Cf. A102283.

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): A319442 ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), this sequence ("mu", A008683).

Equivalent in the ring of Gaussian integers: A318608.

Programs

Mathematica

f[p_, e_] := If[p == 3 || e > 1, 0, Switch[Mod[p, 3], 1, 1, 2, -1]]; eisMu[1] = 1; eisMu[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisMu, 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==3||e>=2, r=0);
        if(Mod(p, 3)==2&e==1, r*=-1);
    );
    return(r);
}

Formula

a(n) = 0 if n is divisible by 3 or has a square prime factor, otherwise Product_{p divides n} (3 - 2*(p mod 3)) where the product is taken over the primes.

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

For squarefree n, a(n) = Legendre symbol (n, 3) = Kronecker symbol (-3, n) = A102283(n).