Type: Article
Publication Date: 2005-04-30
Citations: 13
DOI: https://doi.org/10.4171/rmi/423
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.