1549380232-Automorphic_Forms_and_Applications__Sarnak_

210 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

and hence

~ q - 1 {3 d^2 +1 {3

T1 + T2 = L -

2

• 
  
  
  
  • O(QY^1 - +c) + O(Ql+- 2-y-)
    q~Q
  
  
  
  

which gives Proposition 1.2, on choosing Y = Q d

2

2+

1

• D

Remark 1.14. This method has several advantages over the first proof. Firstly, it
can be adapted to deal with the non-archimedean places as well. Secondly, it can be
extended to automorphic forms over general number fields and provides a bound
of the same quality ed = ~ - d 2 ~ 1 , independently of the base field, while the first

method instead gives a degenerating (although still non-trivial) bound~ - md~+l'
where m is the degree of the base field [LRS2].