The key is the Riemann zeta function. If you take any s > 1, then the series Sum(1/n2,n=1..infinity) converges, and the value of that sum is defined as zeta(s). So zeta is a perfectly well-defined function for s > 1. But it turns out that there's a very powerful and somewhat surprising theorem that if you have a function which is defined on some region, there is at most one function which is smooth on the complex plane and which matches up with that function. This complex function is called the analytic continuation of the original function, and since folks like analytic functions, there's a tendancy to think of the analytic continuation as being the "same" as the original function. So, if we take the analytic continuation of the Riemann zeta function, we can evaluate it anywhere we like, not just for s real and greater than 1. Suppose, for instance, that we evaluate zeta(-1): It happens that that gives us 1/12. But if that's really the same zeta, then from the original definition, zeta(-1) is (or should be) also equal to 1 + 2 + 3 + 4 + ....
Answer A) 30° East longitude (I walked north so I stayed on the same longitude),
10010 km north of the Equator (I'm 20 km further from the Equator).
Answer B) 150° West longitude, 9990 km north of the Equator.
Clearly, these formulae do not make sense if one stays within the traditional way to evaluate infinite series, and so it seems that one is forced to use the somewhat unintuitive analytic continuation interpretation of such sums to make these formulae rigorous. But as it stands, the formulae look "wrong" for several reasons. Most obviously, the summands on the left are all positive, but the right-hand sides can be zero or negative. A little more subtly, the identities do not appear to be consistent with each other...
However, it is possible to interpret (4), (5), (6) by purely real-variable methods, without recourse to complex analysis methods such as analytic continuation, thus giving an "elementary" interpretation of these sums that only requires undergraduate calculus; we will later also explain how this interpretation deals with the apparent inconsistencies pointed out above.
« Older Ars Technica Op-Ed discusses how the Internet of T... | I got thrown out of my first b... Newer »
This thread has been archived and is closed to new comments
Buy a Shirt