A note on the mean value of the zeta and \(L\)-functions. XIV. (English) Zbl 1052.11035

The aim of this note is to develop the study on the feasability of a unified theory of mean values of automorphic \(L\)-functions. This is an important goal, and the present work is an outcome of investigations begun in part XII of the present notes [Proc. Japan Acad., Ser. A 78, 36–41 (2002; Zbl 1106.11305)]. In that work the author laid the framework on the basis of the theory of automorphic representations, and a general approach to the theory of mean values was envisaged.
The author defines the automorphic \(L\)-function associated with the irreducible \(\Gamma\)-automorphic representation \(V\) by \[ L_V(s) = \sum_{n=1}^\infty\rho_V(n)n^{-s}, \] which converges for \(\operatorname{Re} s > 1\) and has analytic continuation to an entire function of polynomial order in any fixed vertical strip. The problem is to evaluate asymptotically the mean square \[ \int_{-\infty}^\infty | L_V({1\over2}+it)| ^2g(t)\,dt, \] where the weight function \(g(t)\) is assumed to be even, entire, real valued if \(t\) is real, and of fast decay in any fixed horizontal strip.
The new approach is the interesting idea to apply suitably the Kirillov map. Because of its geometric nature the author’s method appears to extend to bigger Lie groups. The details and the required notions and notation are to be found in the paper, which is essentially self-contained.


11F70 Representation-theoretic methods; automorphic representations over local and global fields
11M41 Other Dirichlet series and zeta functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations


Zbl 1106.11305
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