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.
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