A103226 Moebius function mu(n+ki) defined for the Gaussian integers. The table begins with n=k=0 and is read by antidiagonals.

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

Offset: 1

Author

T. D. Noe, Jan 26 2005

Keywords

• nice
• sign
• tabl

Comments

The usual definition of the Moebius function is used, except that Gaussian primes are used instead of rational primes. Consider the diagonal (a-b)+bi of Gaussian integers for 0<=b<=a. It appears that the diagonals for a=1, 3, 5 and 11 are the only ones containing just -1 and 1; these Gaussian integers are all squarefree. Interestingly, as shown in A103227, for all n there is some 0<=k<=12 such that n+ki is a squarefull Gaussian integer.

Examples

			The table is symmetric and begins
   0  1  0 -1  0  1  0 -1  0  0  0
   1 -1 -1  1 -1  1 -1  0  1  1 -1
   0 -1  0 -1  0 -1  0 -1  0  1  0
  -1  1 -1  1  0  1  1  1 -1 -1 -1
   0 -1  0  0  0 -1  0  1  0 -1  0
   1  1 -1  1 -1 -1 -1  1 -1  1  0
   0 -1  0  1  0 -1  0  1  0  1  0
  -1  0 -1  1  1  1  1  1 -1 -1 -1
   0  1  0 -1  0 -1  0 -1  0  1  0
   0  1  1 -1 -1  1  1 -1  1  0 -1
   0 -1  0 -1  0  0  0 -1  0 -1  0
		

Links

• T. D. Noe, Plot of the Moebius function for Gaussian Integers
• Eric Weisstein's World of Mathematics, Moebius Function

Crossrefs

Cf. A103227 (least k such that (2n-1)+ki is squarefull).

Programs

Mathematica

moebius[z_] := Module[{f, mu}, If[z==0, mu=0, If[Abs[z]==1, mu=1, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, f=Drop[f, 1]]; mu=1; Do[If[f[[i, 2]]==1, mu=-mu, mu=0], {i, Length[f]}]]]; mu]; Flatten[Table[z=(n-k)+k*I; moebius[z], {n, 0, 15}, {k, 0, n}]]