Euler style – E. Kowalski's blog

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 $d(n)$ (the number of positive divisors of $n$ ) is the Voronoi summation formula, which expresses a sum

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

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

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

where $\bar{a}$ is the inverse of $a$ modulo $c$ , and

$W(y)=\int w(|x|) k(xy)dx$

is some integral transform of $w$ , with kernel $k(y)$ involving the classical Bessel functions $Y_0$ and $K_0$ . Precisely, we have

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

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

$\sum_{n\leq X}d(n),$

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

$w \mapsto W.$

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 $g(x,y)$ of two real variables defined by

$g(x,y)=w(|xy|),$

then it turns out that we have

$\hat{g}(u,v)=W(uv),$

where $\hat{g}$ is the standard Fourier transform of $g$ (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

$\int \int |w(|xy|)|^2dxdy = \int\int |W(uv)|^2dudv.$

Offhandedly, by changing variables, this means that

$\int |w(|t|)|^2 dt \times I = \int |W(s)|^2 ds \times I,$

which would give

$2\|w\|^2= \|W\|^2\quad\quad\quad\quad\quad\quad (\star)$

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

$I=\int \frac{dx}{|x|}.$

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

One can then prove rigorously the formula $(\star)$ 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 $G=\mathrm{SL}_2(\mathbf{R})$ . What happens is that there exists a unitary representation $\rho$ of $G$ (the principal series with Casimir eigenvalue $1/4$ ) which can be represented as acting on the Hilbert space $H=L^2(\mathbf{R},|x|^{-1}dx)$ (the Kirilov model) in such a way that the unitary operator

$T=\rho\Bigl(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\Bigr)$

is given by an integral operator

$(T\varphi)(x)=\int \varphi(y) j(xy)\frac{dy}{|y|}$

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

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

(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

$\int |\varphi(x)|^2 \frac{dx}{|x|}=\int |T(\varphi)(x)|^2\frac{dx}{|x|},$

which holds for all $\varphi\in H$ because $T$ is unitary on $H$ , we deduce exactly $(\star)$ …

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

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

In terms of representation-theory, the story is the same as above, except that the representation $\rho$ 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.

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Kowalski