# extended Moebius transforms, anyone?

Marc LeBrun mlb at fxpt.com
Wed Dec 17 22:29:53 CET 2003

```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

```