{"id":3493,"date":"2012-10-28T14:53:51","date_gmt":"2012-10-28T12:53:51","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=3493"},"modified":"2024-02-20T13:40:35","modified_gmt":"2024-02-20T11:40:35","slug":"euler-style","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2012\/10\/28\/euler-style\/","title":{"rendered":"Euler style"},"content":{"rendered":"

Courtesy of the divisor function, here is another fun example of reasoning in the great style of Euler (the last installment<\/a> is rather old…) A classical tool to study the distribution of values of $latex d(n)$ (the number of positive divisors of $latex n$) is the Voronoi summation formula<\/i><\/a>, which expresses a sum<\/p>\n

$latex S(w,c,a)=\\sum_{n\\geq 1}d(n)w(n)e\\Bigl(\\frac{an}{c}\\Bigr),$<\/p>\n

for a nice test function $latex w$, some positive integer $latex c\\geq 1$, and some integer $latex a$ coprime to $latex c$, in terms of a “dual sum”<\/p>\n

$latex S(W,c,\\bar{a})=\\sum_{m\\in \\mathbf{Z}-\\{0\\}}{d(|m|)W(m\/c^2)e\\Bigl(\\frac{\\bar{a}m}{c}\\Bigr)},$<\/p>\n

where $latex \\bar{a}$ is the inverse of $latex a$ modulo $latex c$, and <\/p>\n

$latex W(y)=\\int w(|x|) k(xy)dx$<\/p>\n

is some integral transform of $latex w$, with kernel $latex k(y)$ involving the classical Bessel functions $latex Y_0$ and $latex K_0$. Precisely, we have<\/p>\n

$latex k(y)=\\begin{cases} -2\\pi Y_0(4\\pi \\sqrt{y})&\\text{ if } x>0\\\\ 4 K_0(4\\pi\\sqrt{|y|})&\\text{ if } y<0\\end{cases},$<\/p>\n

and one should add that there is also a main term in the Voronoi formula, but it is irrelevant for today's story. A classical application of this formula is to improve the error term in Dirichlet's asymptotic evaluation of <\/p>\n

$latex \\sum_{n\\leq X}d(n),$<\/p>\n

which was done indeed by Voronoi.<\/p>\n

In an ongoing work with \u00c9. Fouvry, S. Ganguly and Ph. Michel, we needed to know some unitarity property of the transformation<\/p>\n

$latex w \\mapsto W.$<\/p>\n

This is an entirely classical question, but we didn't find a ready-made statement in Watson’s book on Bessel functions<\/a>. There is however a formal argument that suggests the answer: if we consider the function $latex g(x,y)$ of two real variables defined by<\/p>\n

$latex g(x,y)=w(|xy|),$<\/p>\n

then it turns out that we have<\/p>\n

$latex \\hat{g}(u,v)=W(uv),$<\/p>\n

where $latex \\hat{g}$ is the standard Fourier transform of $latex g$ (this is contained in Section 4.5 of the book of H. Iwaniec and myself.) Hence we have, by the unitarity of the Fourier transform, the identity<\/p>\n

$latex \\int \\int |w(|xy|)|^2dxdy = \\int\\int |W(uv)|^2dudv.$<\/p>\n

Offhandedly, by changing variables, this means that<\/p>\n

$latex \\int |w(|t|)|^2 dt \\times I = \\int |W(s)|^2 ds \\times I,$<\/p>\n

which would give<\/p>\n

$latex 2\\|w\\|^2= \\|W\\|^2\\quad\\quad\\quad\\quad\\quad\\quad (\\star)$<\/p>\n

(the factor $latex 2$ comes from the fact that $latex w$ is extended to an even function on $latex \\mathbf{R}$ from its original source as a function defined for non-negative real numbers), if not for the fact that the “constant” $latex I$ is the integral<\/p>\n

$latex I=\\int \\frac{dx}{|x|}.$<\/p>\n

Alas, it diverges, although probably Euler would write it as $latex I=4\\log (\\infty)$ (two infinities from the divergence at $latex 0^{\\pm}$, the other two from the divergence at $latex \\pm \\infty$), and be happy with the outcome.<\/p>\n

One can then prove rigorously the formula $latex (\\star)$ by truncation arguments, but here is a more conceptual argument (which offers the advantage of being something we can just quote), which follows from the interpretation of the Voronoi formula in terms of the representation theory of $latex G=\\mathrm{SL}_2(\\mathbf{R})$. What happens is that there exists a unitary representation $latex \\rho$ of $latex G$ (the principal series with Casimir eigenvalue $latex 1\/4$) which can be represented as acting on the Hilbert space $latex H=L^2(\\mathbf{R},|x|^{-1}dx)$ (the Kirilov model<\/i>) in such a way that the unitary operator<\/p>\n

$latex T=\\rho\\Bigl(\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix}\\Bigr)$<\/p>\n

is given by an integral operator<\/p>\n

$latex (T\\varphi)(x)=\\int \\varphi(y) j(xy)\\frac{dy}{|y|}$<\/p>\n

for some function $latex j$, which Cogdell and Piatetski-Shapiro called the Bessel function<\/i> of $latex \\rho$ (see this note of Cogdell<\/a> for a short explanation of this, with the analogues for finite fields and $latex p$-adic fields). Now, by direct inspection of the formula for $latex j(y)$ that Cogdell and Piatetski-Shapiro computed, and comparison with the kernel $latex k(y)$ in the Voronoi formula, one finds that<\/p>\n

$latex W(y)=|y|^{-1\/2} T( x\\mapsto \\sqrt{|x|} w(|x|) )$<\/p>\n

(in this other short note<\/a>, Cogdell explains why it is no coincidence that this abstract Bessel function appears in the Voronoi summation formula). Now, from<\/p>\n

$latex \\int |\\varphi(x)|^2 \\frac{dx}{|x|}=\\int |T(\\varphi)(x)|^2\\frac{dx}{|x|},$<\/p>\n

which holds for all $latex \\varphi\\in H$ because $latex T$ is unitary on $latex H$, we deduce exactly $latex (\\star)$… <\/p>\n

Remark.<\/b> There is a completely similar story where the circles $latex x^2+y^2=a$ replace the hyperbolas $latex xy=a$, or in other words, if one defines
\n$latex g(x,y)=w(x^2+y^2).$<\/p>\n

Then the Fourier transform of $latex g$ is still a radial function $latex W(u^2+v^2)$, and the map $latex w\\mapsto W$ is a Hankel transform (it involves the Bessel function $latex J_0$). Its unitarity follows then immediately from that of the Fourier transform, since the analogue of the divergent integral $latex I$ is now, indeed, a finite constant.<\/p>\n

In terms of representation-theory, the story is the same as above, except that the representation $latex \\rho$ is replaced with a discrete series representation. One can also deal similarly with radial functions in higher-dimensional euclidean spaces, which involves other discrete series representations.<\/p>\n","protected":false},"excerpt":{"rendered":"

Courtesy of the divisor function, here is another fun example of reasoning in the great style of Euler (the last installment is rather old…) A classical tool to study the distribution of values of $latex d(n)$ (the number of positive divisors of $latex n$) is the Voronoi summation formula, which expresses a sum $latex S(w,c,a)=\\sum_{n\\geq … Continue reading Euler style<\/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-3493","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/3493","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=3493"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/3493\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=3493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=3493"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=3493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}