Mean value of Dirichlet series associated with holomorphic cusp forms
Let \(F(z)= \sum_{n=1}^ \infty f_ n e(nz)\) be a holomorphic cusp form of weight \(k\) for the full modular group. Suppose also that \(F\) is an eigenfunction for all the Hecke operators, and let \(L_ F (s)= \sum_{n=1}^ \infty f(n) n^{-s}\) with \(f(n)= f_ n n^{(1-k )/2}\) be the associated Dirichlet series. The main result (Theorem 1) gives a ``mollifier'' \(B(s)= \sum_{\alpha \leq B} b_ \alpha \alpha^{-s}\) for \(L_ F (s)\) is the sense that if \(B= T^ \theta\) with \(0<\theta <1/6\), then uniformly for \(| u|+ | v|\ll 1/\log T\) it holds \[ \int_ 1^ T L_ F ({\textstyle {1\over 2}}+ u+it) L_ F({\textstyle {1\over 2}}+ v-it) | B({\textstyle {1\over 2}}+ it)|^ 2 dt\sim \beta T, \] where \(\beta\ll 1\) depends on \(u\), \(v\), and the structure of the mollifier. This is applied to the zeros of the derivative of the function \(\xi_ F (s)= (2\pi )^{-s} \Gamma((k- 1)/2+ s) L_ F(s)\) on the critical line \(\sigma= 1/2\); for instance, at least a proportion 0.326 of the zeros of \(\xi_ F^ \prime (s)\) are simple and lie on the critical line. Conclusions like this -- analogs of a celebrated theorem of A. Selberg on Riemann's zeta-function -- can be drawn by the well-known method of N. Levinson. But this argument fails to cover the function \(L_ F (s)\) itself, for proving that a positive proportion of its zeros lie on the critical line would require a longer mollifier with \(\theta> 0.327\). However, this analog of Selberg's theorem has been established by \textit{J. L. Hafner} [Topics in analytic number theory, 127-164 (1985; Zbl 0603.10026)] following Selberg's original approach. As another main result (Theorem 2), an asymptotic formula for the mean value of \(| L_ F ({1\over 2}+it) A({1\over 2}+it)|^ 2\) is given, where \(A(s)\) is a Dirichlet polynomial of length \(T^ \theta\) with \(\theta< 1/6\). The proofs of the theorems are based on spectral methods.
scientific article; zbMATH DE number 701299