Taille des valeurs de fonctions $L$ de carrés symétriques au bord de la bande critique

Abstract

For each weight k and level N square free and without small prime factors, we prove the existence of primitive forms f_+ and f_- of weight k and level N such that L(1,\operatorname{sym}^2f_+)\gg_{k}[\log\log(3N)]^{3} and L(1,\operatorname{sym}^2f_-)\ll_{k}[\log\log(3N)]^{-1}. The result comes from a delicate study of the moments of L(1,\operatorname{sym}^2 f) . This study gives also results for squarefree levels but with small prime factors. It provides counterexamples to the equivalence between harmonic and natural means.