“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<\/p>\n
$latex \\zeta(2)=\\sum_{n\\geq 1}{\\frac{1}{n^2}},$<\/p>\n
which Euler was the first to evaluate.<\/p>\n
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<\/i><\/a> class.<\/p>\n Consider the Hilbert space<\/p>\n $latex H=L^2([0,1])$<\/p>\n (for the Lebesgue measure), and the Volterra operator<\/a> T<\/i>:<\/p>\n $latex Tf(x)=\\int_0^x{f(y)dy}$<\/p>\n which is a linear operator acting on H<\/i>, in fact a Hilbert-Schmidt operator<\/a> with kernel<\/p>\n $latex k(x,y)=\\mathbf{1}_{\\{y\\leq x\\}}$<\/p>\n which is bounded, and therefore certainly belongs to the space<\/p>\n $latex L^2([0,1]^2,dxdy).$<\/p>\n The operator T<\/i> is therefore compact, and the operator S=T*<\/sup>T<\/i> is also compact, and in fact positive, so that the trace<\/i> is well-defined as a non-negative real number, or infinity. The trace<\/a> is well-known to be expressible in different ways: (i) as a sum of the series formed with the eigenvalues of S<\/i> (with multiplicity); (ii) as the sum of the series<\/p>\n $latex \\sum_{n}{\\langle S(f_n),f_n\\rangle}=\\sum_{n}{||Tf_n||^2},$<\/p>\n for an arbitrary choice of orthonormal basis (fn<\/sub>)_n<\/i> of H<\/i>; (iii) as the integral<\/p>\n $latex \\int_0^1\\int_0^1{|k(x,y)|^2dxdy}.$<\/p>\n This last integral is of course completely elementary: it is the area of the lower-half triangle in the square [0,1]2<\/sup><\/i> below the diagonal; in other words, it is 1\/2.<\/p>\n For an alternate expression (hence an identity), we look at the series above for the easiest orthonormal basis available:<\/p>\n $latex f_n(t)=e^{2i\\pi nt}\\quad\\quad\\text{ for } n\\in\\mathbf{Z}.$<\/p>\n For the special case n=0<\/i>, we have<\/p>\n $latex Tf_0(x)=x,$<\/p>\n hence<\/p>\n $latex \\langle Sf_0,f_0\\rangle =||Tf_0||^2=\\int_0^1{x^2dx}=\\frac{1}{3}.$<\/p>\n For non-zero n<\/i>, we have<\/p>\n $latex Tf_n(x)=\\frac{e^{2i\\pi nx}-1}{2i\\pi n},$<\/p>\n and therefore (Parseval, if you wish, or direct computation):<\/p>\n $latex \\langle Sf_n,f_n\\rangle = ||Tf_n||^2=\\frac{1}{4\\pi^2 n^2}\\times (1+1)=\\frac{1}{2\\pi^2n^2}.$<\/p>\n Summing over all n<\/i> and identifying the two expression for the trace, we get<\/p>\n $latex \\frac{1}{2}=\\frac{1}{3}+\\sum_{n\\not=0}{\\frac{1}{2\\pi^2n^2}}=\\frac{1}{3}+\\frac{1}{\\pi^2}\\zeta(2),$<\/p>\n and hence — unsurprisingly, I presume — we get<\/p>\n $latex \\zeta(2)=\\frac{\\pi^2}{6}.$<\/p>\n (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 \u256c\u00c2(2)=\u00a4\u00c72<\/sup>\/12<\/i>).<\/p>\n As a proof (which, I repeat, is certainly not new, though it is not found in this collection<\/a>), this is fairly close in flavor to the Fourier-expansion proofs, where one expands (typically) the function x-1\/2<\/i> on [0,1]<\/i> into Fourier series before applying the Parseval identity. (In fact, it seems this is “dual” in some simple way which could be made precise).<\/p>\n Like the Fourier-expansion argument, it has the nice feature of showing almost immediately that it will be possible to generalize the argument to<\/p>\n $latex \\zeta(2k)=\\sum_{n\\geq 1}{\\frac{1}{n^{2k}}}$<\/p>\n for k> 0<\/i> integer, using Tk<\/sup><\/i> instead of T<\/i>; it also hints quite strongly that the result will be of the type<\/a><\/p>\n $latex \\zeta(2k)=\\alpha_k\\pi^{2k},$<\/p>\n for some rational number<\/i> \u256c\u2592k<\/sub><\/i>. But it is equally obvious that this will not work at all like this for zeta evaluated at odd positive integers, as it should…<\/p>\n","protected":false},"excerpt":{"rendered":" “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 $latex \\zeta(2)=\\sum_{n\\geq 1}{\\frac{1}{n^2}},$ which Euler was the first to evaluate. It is possible to … Continue reading Euler for another day<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-677","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/677","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=677"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/677\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=677"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=677"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=677"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}