A note on the mean value of the zeta and \(L\)-functions. XV. (English) Zbl 1196.11073

Extended summary: The aim of the present article is to render the spectral theory of mean values of automorphic \(L\)-functions – in a unified fashion: \[ {\mathcal M}(A,g):=\int_ {-\infty}^ {\infty}| L_ A(1/2+it)| ^ 2 g(t)\,dt \] of the automorphic \(L\)-function \(L_ A\), where \(A\) is an irreducible automorphic representation for the full modular group, and \(g\) is a suitable weight function. The spectral decomposition is given in the form \[ \begin{split} {\mathcal M}(A,g)=m(A,g)+\\ +2\text{Re}\,\left\{\sum_ V L_ V(1/2)\Theta_ A(V;g)+\int_ {(0)}\frac{\zeta(1/2+\nu) \zeta(1/2-\nu)}{\zeta(1+2\nu)\zeta(1-2\nu)} R_ A(1/2+\nu)\Theta_ A(\nu;g)\frac{d\nu}{4\pi i}\right\},\end{split} \] with leading term \(m(A;g)\).
This is an outcome of the investigation commenced with the parts XII [Proc. Japan Acad., Ser. A 78, No. 3, 36–41 (2002; Zbl 1106.11305)] and XIV [Proc. Japan Acad., Ser. A 80, No. 4, 28–33 (2004; Zbl 1052.11035)], where a framework was laid on the basis of the theory of automorphic representations and a general approach to the mean values was envisaged. We restrict ourselves to the situation offered by the full modular group, solely for the sake of simplicity. Details and extensions are to be published elsewhere.


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11M41 Other Dirichlet series and zeta functions
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