Integral mean values of modular \(L\)-functions
The author considers the mean squares of \(L\)-functions associated to modular forms with respect to Hecke congruence subgroups, expressing the mean value as an inner product. This avoids the discussion of generalized additive divisor problems. More precisely, let \(f\in\mathcal S_k(N, \chi)\) be a cusp form for \(\Gamma_0(N)\) of weight \(k\) and character \(\chi\bmod N\). Let \(L(s, f)\) be associated \(L\)-function, normalized so that the central point is at \(s=1/2\). The author proves that \[ \int_0^T\left| L\left(\frac12+it, f\right)\right| ^2\,dt\sim\frac{2(4\pi)^k}{\Gamma(k)} \frac{\| f\| ^2}{\text{vol}(\Gamma_0(N)\backslash\mathfrak{H})}\,T\log T, \] where the norm on \(\mathcal S_k(N, \chi)\) is induced from the inner product \[ \langle f_1, f_2\rangle=\iint_{\Gamma_0(N)\backslash\mathfrak{H}} f_1(z)\overline{f_2(z)}y^k\,\frac{dx\,dy}{y^2}. \] An asymptotic formula for weighted mean square is also obtained. For the proof he considers the Dirichlet integral \[ Z_f(w)=\int_1^\infty\left| L\left(\tfrac12+it, f\right)\right| ^2t^{-w}\,dt \] and studies its analytic properties, especially its meromorphic continuation beyond \(\Re w=1\) and its polar behavior. The author points that this realizes and generalizes the ideas suggested by \textit{A. Good} [``The Selberg trace formula and related topics'', Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 53, 207--214 (1986; Zbl 0601.10020)].
scientific article; zbMATH DE number 5009840
scientific article