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 $$ L(1,\sym^2f_+)\gg_{k}[\log\log(3N)]^{3} $$ and $$ L(1,\sym^2f_-)\ll_{k}[\log\log(3N)]^{-1}. $$ The result comes from a delicate study of the moments of $L(1,\sym^2 f)$. This study gives also results for squarefree levels but with small prime factors. It provides counterexamples to the equivalence between harmonic and natural means.