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 ,
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