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
$latex S(w,c,a)=\sum_{n\geq 1}d(n)w(n)e\Bigl(\frac{an}{c}\Bigr),$
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”
$latex S(W,c,\bar{a})=\sum_{m\in \mathbf{Z}-\{0\}}{d(|m|)W(m/c^2)e\Bigl(\frac{\bar{a}m}{c}\Bigr)},$
where $latex \bar{a}$ is the inverse of $latex a$ modulo $latex c$, and
$latex W(y)=\int w(|x|) k(xy)dx$
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
$latex k(y)=\begin{cases} -2\pi Y_0(4\pi \sqrt{y})&\text{ if } x>0\\ 4 K_0(4\pi\sqrt{|y|})&\text{ if } y<0\end{cases},$
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
$latex \sum_{n\leq X}d(n),$
which was done indeed by Voronoi.
In an ongoing work with É. Fouvry, S. Ganguly and Ph. Michel, we needed to know some unitarity property of the transformation
$latex w \mapsto W.$
This is an entirely classical question, but we didn't find a ready-made statement in Watson’s book on Bessel functions. 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
$latex g(x,y)=w(|xy|),$
then it turns out that we have
$latex \hat{g}(u,v)=W(uv),$
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
$latex \int \int |w(|xy|)|^2dxdy = \int\int |W(uv)|^2dudv.$
Offhandedly, by changing variables, this means that
$latex \int |w(|t|)|^2 dt \times I = \int |W(s)|^2 ds \times I,$
which would give
$latex 2\|w\|^2= \|W\|^2\quad\quad\quad\quad\quad\quad (\star)$
(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
$latex I=\int \frac{dx}{|x|}.$
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.
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) in such a way that the unitary operator
$latex T=\rho\Bigl(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\Bigr)$
is given by an integral operator
$latex (T\varphi)(x)=\int \varphi(y) j(xy)\frac{dy}{|y|}$
for some function $latex j$, which Cogdell and Piatetski-Shapiro called the Bessel function of $latex \rho$ (see this note of Cogdell 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
$latex W(y)=|y|^{-1/2} T( x\mapsto \sqrt{|x|} w(|x|) )$
(in this other short note, Cogdell explains why it is no coincidence that this abstract Bessel function appears in the Voronoi summation formula). Now, from
$latex \int |\varphi(x)|^2 \frac{dx}{|x|}=\int |T(\varphi)(x)|^2\frac{dx}{|x|},$
which holds for all $latex \varphi\in H$ because $latex T$ is unitary on $latex H$, we deduce exactly $latex (\star)$…
Remark. 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
$latex g(x,y)=w(x^2+y^2).$
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.
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.
