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).


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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