8 STEVEN ZELDITCH

(0.7) Theorem (e.g. [V, Theorem 5.2.1])

Mr(T) +Nr(T)

= 2

1 |F| T2 + 0(T InT),

where |F| is the area of a fundamental domain F for T.

The proof of the analogues of (0.7) for a ± 1 first involves the sum

function:

(0.8) Definition Nr(r,T) = S 0p(r) u., u-.

Here, the matrix coefficients 0p(r) u., u- are well defined for any

automorphic form a (cusp form, Eisenstein series or residue) due to the rapid

decay of u. in the cusps. The continuous analogue Mp(r,T) is harder to

define. When a is a, cusp from, the matrix coefficients 0p(j) E(-,^+ir),

E(-,w+ir) are well-defined and continuous in r, and we may set

(0.9) Definition Mr(r,T) = - ^ [ 0p(r) E(-,j+ir), E(-,j+ir) dr

(for cuspidal a).

Vhen a is an Eisenstein series, the matrix coefficient is not well-defined.

Ve will follow Zagier's method of renormalizing inner products [Za 2] to give

it a meaning . Ve therefore replace 0p(r) E(-,j+ir), E(-,j+ir) by its

renormalization RN0p(r) E(«,j+ir), E(-,j+ir), whose precise definition and

properties will be discussed in §4. For the moment, we only note that, if a =

1

E(-,s), it coincides with a Rankin-Selberg convolution

R(|E(-,^+ir)|2,s)

(cf.