Elstrodt , Grunewald and Mennicke, Eisenstein series

181

Proof . We correct a sign error in the Kronecker limit formula in the version of Lang [26], p. 280, (2) and obtain

C ( m , u, H-j) = -^(- + 2y-logHj,|--logg(mu-i) + 0(5)) for s^Q.

This implies

( Nuy + 4 ( m , u , l + 5)r^+*

= -5=^r(- + 2}'-log|4l+log(rNu)--logg(mu-i) + 0(5)) for s^O. In addition, we have from the functional equation of the zeta-function (see [25], p. 254)

2n

\dg\s

C ( m , u \s)-=

In уГ{\-8)

ш ) ^(1+^)

C ( u Sm, 1-j)

^ , {Inf \ \ + ys + - l+5log^+- •

^ ( - - + 2y-\og\d^\-Uogg{m-'vi-') + 0{s)

0 / i 1

- Д=г ( - - - - - - 21og27c - - logg ( m - ^u - ^ ) - hO ( 5 ) 1 for 5->0.

/ Щ\ s 6

This yields

In

1 o\ Д-s

\d^\s

Nu^ ''CCîît, u ,'У) ^

: ^^^r ( - - + log(rNu)-21og27r-^logg(m-^u-^) + 0(5)) for 5-^0,

and the assertion follows from (4. 22). П

5 . 3 Theorem. Let the notation be as in Theorem 5. 2. Then

( ^ 4n^ 1

5 - 1 V \df.\ s-i

j=|o " |C ( ( m - ^u ) * , 2 ) Nu2r2

4n^ ( 1

+ ^jT|27-l-log|4|-log(rNu)--logg(mu)

/ 4я|С0| r\ 1ш(Щ^,г\\

+ 2Nu E |co|(7_,(m,u,co)rJJ:J "L e ^K^' ^

0 * CO€U2 \ y\d^ ) )