SELBERG TRACE FORMULA

9

A J 1

(A13(c))); while if a = 1, it coincides with j~(j+ir). Thus,

(0.10) Definition Mp(r,T) = - ^ [ RN0p(r) E(.,^+ir), E(.,j+ir) dr.

The analogues of the Veyl law for cusp forms is:

(0.11) Theorem (4A)(Mp + Np)(r,T) = 0(T/ln T) (a cuspidal).

The proof of (0.11) is very similar to that in the case of co-compact Y ([Z.2,

§4])-

The Veyl law for Eisenstein series a = E(-,s) is quite different. Using

[D-I], we show:

(0.12) Theorem (4B) (Mp + Np)(E(.,s), T) = Q(T3/2) Res = 1/2.

The estimate of (0.12) is of some interest in its own right, since it gives a

partial confirmation of the mean Lindelof hypothesis (MLH) for Rankin-Selberg

zeta functions ([1.1, p.139; 1.2, p.188).

(0.13) Conjecture (MLH) For Y = SL2(Z) and Res = 1/2,

E |Eu., u.| « |s|AT3/2+e

|rdlT

For T = SLo(ff), the Mp term is known to be of lower order than Np, so that

(0.13) is just the conjecture that the estimate in (0.12) holds for the

associated absolute sum. The transition from (0.12) to (0.13) seems quite