2 A. KNIGHTLY AND C. LI
describable in terms of the Eisenstein series
E(s, z) =
1
2
c,d∈Z
(c,d)=1
y1/2+s
|cz + d|1+2s
(Re(s)
1
2
, y 0,z = x + iy).
More accurately, it involves the analytic continuation to s on the imaginary line.
This analytic continuation is provided by the Fourier expansion
E(s, z) =
y1/2+s
+
y1/2−s

π Γ(s)ζ(2s)
Γ(1/2 + s)ζ(1 + 2s)
(1.4)
+
2y1/2π1/2+s
Γ(1/2 + s)ζ(1 + 2s)
m=0
σ2s(m)|m|sKs(2π|m|y)e2πimx.
Here σ2s(m) =

0d|m
d2s
is the divisor sum, and Ks is the K-Bessel function.
The continuous contribution to the Kuznetsov/Bruggeman formula is the following
integral of the product of two Fourier coefficients of E(it, z) against the function
h(t):
(1.5)
1
π

−∞
(m/n)itσ2it(m)σ2it(n)
|ζ(1 + 2it)|2
h(t)dt.
The Fourier trace formula is then the equality between the sum of (1.3) and (1.5)
on the so-called spectral side, with the geometric side given by
(1.6)
δm,n
π2

−∞
h(t) tanh(πt) t dt +
2i
π
c∈Z+
S(m, n; c)
c

−∞
J2it(


mn
c
)
h(t) t
cosh(πt)
dt.
Using this together with the Weil bound (9.2), Kuznetsov proved a mean-square
estimate for the Fourier coefficients an(uj) ([Ku], Theorem 6), which immediately
implies the bound
an(uj)
j,ε
n1/4+ε
in the direction of the (still open) Ramanujan conjecture an(uj) = O(nε). (See also
[Brug], §4.) This extended Selberg’s result (1.2) to the case of Maass forms.
Kuznetsov also “inverted” the formula to give a variant in which a general test
function appears on the geometric side in place of the Bessel integral. (Motohashi
has given an interesting conceptual explanation of this, showing that the procedure
is reversible, [Mo2].) This allows for important applications to bounding sums of
Kloosterman sums. Namely, Kuznetsov proved that the estimate
(1.7)
c≤X
S(m, n; c)
c
m,n,ε
Xθ+ε
holds with θ =
1
6
([Ku], Theorem 3). The Weil bound alone yields only θ =
1
2
, show-
ing that Kuznetsov’s method detects considerable cancellation among the Kloost-
erman sums due to the oscillations in their arguments. Linnik had conjectured in
1962 that (1.7) holds with θ = 0, and Selberg remarked that this would imply the
Ramanujan-Petersson conjecture for holomorphic cusp forms of level 1, ([Sel3]; see
also §4 of [Mu]). By studying the Dirichlet series
Z(s, m, n) =
c
S(m, n; c)
c2s
,
Previous Page Next Page