One excellent reason to believe that these Cauchy-divergent sums can be assigned reasonable values comes from the fact that equations like
have real, finite combinatorial consequences. These are the sums of the Catalan numbers, Motzkin numbers, and Schroder numbers, respectively. By taking these divergent sums seriously, we are led to new results. As a matter of fact, a new combinatorial theorem came out of the comments in the last post, thanks to Isabel (of God Plays Dice) noting that the sum of the Motzkin numbers should be : there is a bijective algorithm (explicitly constructed) which converts Motzkin trees to 5-tuples of Motzkin trees.
Today I would like to pose the question “How do we know that our divergent sums are meaningful in situations where we can’t immediately find a finite consequence?” And, of course, finally get to the promised puzzle.
By being particularly uncritical about what universe our calculations were living in, we showed last time that the formula quite indirectly implies that
But unlike the case of tree-counting, where the divergent sums lead us to very hands-on combinatorial truths, it is unclear how much faith we should have in this sum. I’ve heard that this sum comes up (and really is -1/12) when computing vacuum expectation values in quantum field theories, but I don’t really know enough to say anything reasonable about this. Hopefully some kind commenter will fill in the blanks. While a physical manifestation of this divergent sum is about the best thing we could hope for, we can also look for other abstract manifestations and see if the value -1/12 is consistent between them.
Remember that when we tried to compute the sum directly using analytic regularization, there was a problem:
which is singular at and therefore fails to give a finite value to this sum. In order to compute the value we had to start from and (both proved by analytic regularization) and then carry out some rather questionable algebraic manipulations to get the final result. Why should we have faith in the answer we computed?
If we believe in a Platonic realm of divergent series, we could think of our value as an experimental prediction: if we find another way to compute divergent sums which assigns a value to , that value will be . This is probably not going to actually be true, but if we had another regularization scheme which did give us , we might be more inclined to believe that the sum has some meaningful finite interpretation just like the combinatorial sums from before.
With that setup, you probably aren’t going to be surprised that we do have another useful regularization scheme for divergent sums. It goes by the name of zeta regularization, and works like this: the zeta-regularized sum is computed by taking the limit as of
Zeta regularization works well in many cases where analytic regularization does not, and vice versa. If there is a universal method for summing divergent series, it is almost as if zeta and analytic regularizations are two disjoint approximations of this method. In particular, we have no reason to expect zeta-regularized sums to have anything to do with analytically regularized sums. With our expectations sufficiently lowered, let us do some calculating with the sum :
where is the Riemann zeta function. The sum we are trying to compute is therefore given by , which we can compute using the functional equation
and the fact (demonstrated nicely by Jim on this very blog) that . What do we get?
Let us pause for a moment and think about just how bizarre this is: two entirely different methods of assigning a sum to the series, the first of which used calculations which are not even clearly well-defined, have given us the same result. Lest we think this is a coincidence, let us also compute with zeta-regularization. Using our questionable algebra from last time, we found a sum of for this series. With zeta-regularization:
so we need to compute . The functional equation tells us that for an infinitessimal ,
(where = should be read as “is infinitessimally close to”). is the harmonic series, which gives a simple pole of reside 1 at . As a result,
Do you believe that there is some rigorous notion of “divergent sum” hiding away in some Platonic corner of the universe yet?
Now here is my puzzle: the harmonic series obviously diverges in the Cauchy sense. It also is Cauchy-divergent for any p-adic metric on . Contrast this property with the nice divergent series , for example. The harmonic series does not have a nice zeta regularization due to the pole of at . It does not have an analytic regularization either:
Sending to 1 is a disaster, so the harmonic series diverges under analytic regularization as well. Unlike all the other divergent series that we have seen so far, the harmonic series seems to be really divergent. This is my puzzle to you, the internet: can you sum the harmonic series?
Just for reference, here are two other dirty tricks that I have tried: the first uses the fact that the alternating harmonic series converges. We have
But as , this becomes the unfortunate equation .
The second trick is much dirtier, and I was very sad to see that it seems to be failing. The zeta function has a special relationship
with the Mobius function. The Mobius function is zero about a third of the time, and is equal to +1 as often as it is equal to -1. So it is not unreasonable to expect that
converges. But numerical tests that I have run computing the sum out to 100 million terms show that , computed this way, is roughly half the magnitude of the nth partial sum of the harmonic series. For reference, if we replace with a random variable that has the same distribution, the expected absolute value of that we obtain is something like 1.8, while the value computed using the real function is about 8.9 and the partial sum of the harmonic series is about 19 after 100 million terms.
Thus concludes my sad story about trying to sum the harmonic series; can we come up with a more clever idea?