1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 5. SOME APPLICATIONS OF SUBCONVEXITY 271

in terms of the class number h( -n) of primitive positive binary quadratic forms

with discriminant -n [A, AB], namely

{

12h(-n) if n = 1, 2(mod 4)

r3(n) = 8h(-n) if n = 3(mod 8)

0 otherwise.

By the class number formula, Siegel's theorem, and the (more elementary) compu-

tation of class numbers of non maximal orders in imaginary quadratic fields (see

[Cor] for instance), it follows that

r3 (n) » 0 n^1 /^2 -^0 if r3 (n) > 0.

Hence, for the unrestricted representations one has

(5 .3) r 3(n) » 0 n^1 /^2 -^0 if n ¢. 0, 4, 7(mod 8).

Thus if n ¢. 0, 4, 7(mod 8), there are many vectors in R 3 (n); one may then look at
the distribution of their projections on the unit sphere S^2 as n --; +oo. This question
was studied by Linnik: by using ergodic methods, he prove the equidistribution of
the projections under the extra assumption that -n is a quadratic residue modulo
some fixed odd prime [Lin3]. In [11], Iwaniec removed this extraneous condition
by using quite different techniques^1.

Theorem 5.2. As n goes to +oo through integers n ¢. 0, 4, 7(mod 8), the set
R 3 ( n) /fa becomes equidistributed on the unit sphere S^2 with respect to the standard
Lebesgue measure; i.e. for any continuous function V on S^2

(5.4)

Proof. By Weyl's equidistribution criterion, it is sufficient to show that for any har-
monic polynomial Pon R^3 of degree k ~ 1, say, the Weyl sum

W(n, P) := r (n)^1 L P( x 1n) --;^1

(^3) x E R 3 (n) y ,. S^2 P(u)dμ = 0.
Observe that W(n, P) = 0 if k is odd. If k is even, one has
...;n- k/2 ...;n-k/2
W(n,P) = rn(n) L P(X.) = rn(n) rp(n)
(^3) xER3 (n) 3
say. In view of (5.3), it is sufficient to show that
k+l {J
rp(n) = O(n- 2 - )
for some absolute 8 > 0. The theta series
8p(z ) := L r p (n)e (n z )
n~O
(^1) This method gives, in fact, an estimate for the speed of convergence in (5.4), which apparently is not
accessible by the ergodic approach.