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(

4π

√

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

,