Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site brl-tgr.ARPA
Path: utzoo!watmath!clyde!bonnie!akgua!gatech!seismo!brl-tgr!gwyn
From: gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>)
Newsgroups: net.math
Subject: Re: Euler's formula and arithmetic dimensions
Message-ID: <64@brl-tgr.ARPA>
Date: Mon, 25-Nov-85 23:16:19 EST
Article-I.D.: brl-tgr.64
Posted: Mon Nov 25 23:16:19 1985
Date-Received: Wed, 27-Nov-85 05:43:00 EST
References: <1095@enea.UUCP>
Organization: Ballistic Research Lab
Lines: 25

>         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".

This formula doesn't even talk about volume, except to the extent
that every number can be thought of as the volume of SOMEthing.

>  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?
> ...
> What Manin proposes above is still more abstract (and not yet used in 
> physics).  The points in R x Spec(Z) would be pairs (a, p) where a is
> an ordinary number and p is a prime number.  There'd also be special
> points (a, 0) that sort of glue the whole thing together.

Tell us more about Spec(Z), whose elements are apparently just the
positive prime integers.  What is this "algebro-geometric spectrum"?

I have to object to calling every factor algebra a "dimension".