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The spectral growth of automorphic $$L$$-functions. (English) Zbl 0746.11024
For the zeta-function of Hecke $${\mathcal H}(s)$$ associated with a Maass cusp form $$u(z)$$ for the modular group which is an eigenfunction of the Laplace operator with eigenvalue $$\lambda=1/4+r^ 2$$, $$r>0$$ is given an estimate on the critical line $$\hbox{Re }s=1/2$$, $${\mathcal H}(s)\ll| s| r^{1/3+\varepsilon}$$, subject to an assumption about the average size of the Fourier coefficients of $$u(z)$$. This upper bound for $${\mathcal H}(s)$$ follows from a sharp estimate for the spectral mean-value of certain linear forms in the Fourier coefficients of $$u(z)$$ which is the main result of the paper (unconditional).

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11F30 Fourier coefficients of automorphic forms
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