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28.10.2012

Euler style

Filed under: Mathematics @ 14:53

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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