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

##### MSC:
 11M41 Other Dirichlet series and zeta functions
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##### References:
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