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From: bill@milford.UUCP (bill)
Newsgroups: net.math
Subject: Re: Euler's formula and arithmetic dimensions
Message-ID: <111@milford.UUCP>
Date: Wed, 27-Nov-85 08:37:31 EST
Article-I.D.: milford.111
Posted: Wed Nov 27 08:37:31 1985
Date-Received: Fri, 29-Nov-85 10:55:03 EST
References: <1095@enea.UUCP>
Organization: Telecomp,Inc. , Milford Ct.
Lines: 38

> < line eater bait >
> 
> In the November issue of Science '85 Yu.I.Manin writes (p.91) that
> Euler's formula
> 
>         6/PI^2 = (1-1/2^2)*(1-1/3^2)*(1-1/5^2)*...
> 
> (2,3,5,... = prime numbers, ^ = exponentiation) "gives the volume of a
> simple solid in a space having one ordinary dimension and one arithmetic
> dimension".  It sounds like he's talking about the space
> 
> 	R x Spec(Z)   (R = reals, Z = integers)
> 
> where Spec(Z) is the algebro-geometric spectrum of integers.  Does
> anybody know more about this?
> 

In the "Ring of Arithmetic Functions" multiplication is defined as

(f * g)(n) = (sum over the divisors of n) of (f(n/d) g(d))

The function e(n) = {1 if n=1; 0 otherwise} turns out to be the identity.

The Moebius function	   1 if n = 1;
		mu(n) = {  0 if n has a square dividing it
			   (-1)^q where q is the number of distinct prime
								divisors

This Moebius function then turns out to be the inverse of the constant 1.

Further  the (infinite sum over n) of (mu(n)/n*n) = 1/zeta(2) = 6/pi*pi.

I'd never think of using convolution multiplication for dimensions, but...

More of interest to me would be questions like
What is the inverse of arithmetic functions like the Euler phi function?

I'm not positive if this is what Manin meant (is he on the net? :-))