“Every sum is a trace” is a well-known folklore saying in certain automorphic circles (echoed by a no less convincing “Every sum is an expectation” around probabilists); in this spirit, let’s have a look at one of the most famous sums

which Euler was the first to evaluate.

It is possible to see it, and then compute it, as a trace, and I’m sure this has been done many times; here is (a variant of) the way I presented things for an exercise for my *Spectral Theory* class.

Consider the Hilbert space

(for the Lebesgue measure), and the Volterra operator *T*:

which is a linear operator acting on *H*, in fact a Hilbert-Schmidt operator with kernel

which is bounded, and therefore certainly belongs to the space

The operator *T* is therefore compact, and the operator *S=T ^{*}T* is also compact, and in fact positive, so that the

*trace*is well-defined as a non-negative real number, or infinity. The trace is well-known to be expressible in different ways: (i) as a sum of the series formed with the eigenvalues of

*S*(with multiplicity); (ii) as the sum of the series

for an arbitrary choice of orthonormal basis *(f _{n})_n* of

*H*; (iii) as the integral

This last integral is of course completely elementary: it is the area of the lower-half triangle in the square *[0,1] ^{2}* below the diagonal; in other words, it is 1/2.

For an alternate expression (hence an identity), we look at the series above for the easiest orthonormal basis available:

For the special case *n=0*, we have

hence

For non-zero *n*, we have

and therefore (Parseval, if you wish, or direct computation):

Summing over all *n* and identifying the two expression for the trace, we get

and hence — unsurprisingly, I presume — we get

(I said unsurprisingly, but I first managed to get confused enough about the computation — for a slightly different operator — that, for a while, I almost convinced myself that *ζ(2)=π ^{2}/12*).

As a proof (which, I repeat, is certainly not new, though it is not found in this collection), this is fairly close in flavor to the Fourier-expansion proofs, where one expands (typically) the function *x-1/2* on *[0,1]* into Fourier series before applying the Parseval identity. (In fact, it seems this is “dual” in some simple way which could be made precise).

Like the Fourier-expansion argument, it has the nice feature of showing almost immediately that it will be possible to generalize the argument to

for *k> 0* integer, using *T ^{k}* instead of

*T*; it also hints quite strongly that the result will be of the type

for some *rational number* *α _{k}*. But it is equally obvious that this will not work at all like this for zeta evaluated at odd positive integers, as it should…

This is quite a beautiful way to compute zeta(2).

This is quite interesting! In fact I am wondering: could those mantras “every sum is a trace” and “every sum is an expectation” be turned into general useful Tricks-wiki articles, like your one on smooth sums?

#2: You’re right that some of these slogans are good potential subjects for Tricki articles, and it wouldn’t be difficult to get many more examples to illustrate them. I may try to do this when I have some time in the next few days.