extended Moebius transforms, anyone?

[Marc LeBrun]

I've recently stumbled onto a possible class of mutant Moebius transforms 
that might prove interesting.

Alas, I'm unable to take it further just now; perhaps someone with sequence 
transform power tools might wish to pursue this?

Consider extending the integers by adjoining, say, sqrt2.  This changes the 
set of primes in the following ways:
   2 is no longer prime, but is now the square of the prime sqrt2
   some of the rational primes (namely those of the form 8k+/-1) can now be 
factored into conjugate primes (a +/- b sqrt2).

Also, any multiplicative integer sequence is also multiplicative in the 
extension (and vice versa, in a natural sense) because the new irrational 
primes must always occur in matched conjugate pairs in any integer's 
factorization.

So we can define an analog of Moebius mu for this extension, call it mu2, 
in a natural way, being the same as mu, except:
   mu2[2k]=0, since 2 is now a square
   primes of the form 8k+/-1 now contribute +1 instead of -1 to the 
products (-1 for each conjugate prime factor)

I've submitted mu2 as A091069 (see below).  It no doubt figures in a lot of 
analog relationships for the sqrt2 extension arithmetic (divisors, zetas etc).

Specifically, can we use this to define an extended analog of the Moebius 
mu transform?  What interesting new sequences or relationships between 
known sequences might it produce?

(This seems plausibly viable, since the mu definition "untangles" 
convolutions over "divisors" in an essentially combinatorial structural 
way, largely independent of what the underlying arithmetic semantics 
actually is).

One detail is what extended values to assign to the transform argument 
sequence at the new irrational divisors.  One way is to simply define 
a(x)=0 for non-integer x, which keeps then from contributing anything to 
the transform sums (another way to suppress them might be to have the value 
at the conjugate of x be -a(x), etc).

Of course the same questions apply for other values of sqrt2.

"Enjoy"!

=========

%I A091069
%S A091069 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
%N A091069 Moebius mu sequence for real quadratic extension sqrt2
%C A091069 Analog of Moebius mu with sqrt2 adjoined. Same as mu (A008683) 
except: 0 for even n (A005843) due to square (extended prime) factor 
(sqrt2)^2, and rational primes of the form 8k+/-1 (A001132) factor into 
conjugate (extended prime) pairs (a + b sqrt2)(a - b sqrt2), thus 
contributing +1 to the product instead of -1; eg 7 = (3+sqrt2)(3-sqrt2).
%D A091069 G. H. Hardy and E. M. Wright, An Introduction to the Theory of 
Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 256, p. 221.
%F A091069 Zero if n even or has a square prime factor, otherwise product 
2-|p mod 8| for each prime p dividing n (ie +1 if p=8k+/-1, -1 if p=8k+/-3).
%e A091069 a(21) = (-1)*(+1) = -1 because 21=3*7 which are respectively +3 
and -1 mod 8
%Y A091069 A008683, A005843, A001132
%O A091069 1
%K A091069 ,mult,easy,sign,
%A A091069 Marc LeBrun (mlb at well.com), Dec 17 2003