On the structure of the Selberg class. VII: \(1<d<2\). (English) Zbl 1235.11085

Kaczorowski, Jerzy; Perelli, Alberto

Ann. Math. (2) 173, No. 3, 1397-1441 (2011).

The paper is devoted to the investigation of the structure of the Selberg class \(S\). The classification of elements from \(S\) is based on a real-valued degree \(d\), and, for every \(L\)-function in \(S\), it is conjectured that \(d \in \mathbb N\). In this paper, the authors prove that, for \(1<d<2\), the subclass \(S^{\#}_d\) of the extended Selberg class is empty, when the subclass \(S^{\#}_d\) consists of the non-zero functions \(F\) with given degree \(d\) and satisfying some additional conditions. The main tool for the proof is a general transformation formula for multidimensional nonlinear twists of functions \(F \in S^{\#}_d\) associated with the function \(f(\xi,{\pmb{\alpha}})\) given by \[ F(s;f)=\sum_{n=1}^{\infty}\frac{a_F(n)}{n^s}e^{-2 \pi i f(n,{\pmb \alpha})}, \quad \sigma>1, \] Here the Dirichlet coefficients \(a_F(n)\) of \(F(s)\) satisfy \(a_F(n) \ll n^\varepsilon\) for every \(\varepsilon >0\), and, for \(\xi>0\), \[ f(\xi,{\pmb \alpha})=\xi^{\kappa_0}\sum_{\nu=0}^{N}\alpha_\nu \xi^{-\omega_\nu} \] with integers \(d \geq 1\) and \(N\geq0\), \({\pmb \alpha}=(\alpha_0,\dots ,\alpha_N)\in \mathbb R^{N+1}\), and \(d\kappa_0>1\), \(0=\omega_0<\dots <\omega_N<\kappa_0\), \(\alpha_0>0\).
For previous parts, see Zbl 1034.11051 (V), Zbl 1082.11055 (VI).

Reviewer: Roma Kačinskaitė (Siauliai)

MSC:

11M41

Other Dirichlet series and zeta functions

Keywords:

converse theorems; degree conjecture; \(L\)-functions; Selberg class

Citations:

Zbl 1034.11051; Zbl 1082.11055

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