Geodesic
On the structure of the Selberg class, VII: $1\lt d\lt 2$
Jerzy Kaczorowski1 ; Alberto Perelli2
1 Faculty of Mathematics and Computer Science<br/>Adam Mickiewicz University<br/> 61-614 Poznań <br/>Poland
2 Dipartimento de Matematica<br/>Università di Genova<br/> 16146 Genova<br/>Italy
Annals of mathematics, Tome 173 (2011) no. 3, pp. 1397-1441.

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The Selberg class $\mathcal{S}$ is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global $L$-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of $\mathcal{S}$. Such a classification is based on a real-valued invariant $d$ called degree, and the degree conjecture asserts that $d\in\mathbb{N}$ for every $L$-function in $\mathcal{S}$. The degree conjecture has been proved for $d\lt 5/3$, and in this paper we extend its validity to $d\lt 2$. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the $L$-functions in $\mathcal{S}$.
DOI : 10.4007/annals.2011.173.3.4

Jerzy Kaczorowski 1 ; Alberto Perelli 2

1 Faculty of Mathematics and Computer Science<br/>Adam Mickiewicz University<br/> 61-614 Poznań <br/>Poland
2 Dipartimento de Matematica<br/>Università di Genova<br/> 16146 Genova<br/>Italy 
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Jerzy Kaczorowski; Alberto Perelli. On the structure of the Selberg class, VII: $1\lt d\lt 2$. Annals of mathematics, Tome 173 (2011) no. 3, pp. 1397-1441. doi : 10.4007/annals.2011.173.3.4. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.3.4/

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