Measure of "adeles minus ideles" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:13:44Z http://mathoverflow.net/feeds/question/49811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49811/measure-of-adeles-minus-ideles Measure of "adeles minus ideles" unknown (google) 2010-12-18T15:41:06Z 2010-12-20T08:54:41Z <p>Hi, I am interested in the set $\mathbb A-\mathbb A^\times$ i.e. the complement of ideles in the adele ring of a number field. </p> <p>Is it measurable, and what is its volume, with respect to the standard measure of adeles?<br> ("standard" means the same as in Tate's thesis) </p> <p>Thank you.</p> http://mathoverflow.net/questions/49811/measure-of-adeles-minus-ideles/49941#49941 Answer by S. Carnahan for Measure of "adeles minus ideles" S. Carnahan 2010-12-20T08:54:41Z 2010-12-20T08:54:41Z <p>This is a bookkeeping post, since the answer seems to have been resolved in the comments. Somebody please vote this up once so this question leaves the "unanswered" queue.</p> <p>Shenghao's answer is essentially that you can view the ideles as a countable union of translates of $\widehat{\mathbb{Z}}^\times$, which has measure zero, so the ideles have measure zero. The measure zero property of $\widehat{\mathbb{Z}}^\times = \prod_p \mathbb{Z}_p^\times$ arises from the fact that for each prime $p$, the set $\mathbb{Z}_p^\times$ has volume $\frac{p-1}{p}$ in $\mathbb{Z}_p$, and the product of these numbers over primes $p$ absolutely converges to zero. Very loosely speaking, we have $\prod_p \frac{p-1}{p} = \frac{1}{\zeta(1)}$ and the latter term is morally zero, since it is the reciprocal of the harmonic series.</p> <p>In conclusion, the adeles minus the ideles is a measurable set with infinite volume, because the volume of the adeles is infinite and the measure of the ideles is zero.</p> <p>Apparently, this fact was a primary flaw in a serious attempt at a proof of the Riemann Hypothesis from a couple years ago. The mathematician in question tried to manipulate the integral (over the adeles) of a function supported on the ideles as if it were nonzero.</p>