On zeta-functions associated to certain cusp forms. I
Let \(F(z)\) be a holomorphic cusp form of weight \(\kappa\) for the full modular group which is also assumed to be a normalized eigenform for all Hecke operators. Then there is the Fourier expansion \(F(z)=\sum_{m=1}^{\infty}c(m)e^{2\pi imz}\) with \(c(1)=1\). The associated zeta function is \(\phi(s,F)=\sum_{m=1}^{\infty}{c(m)\over m^{s}}\). This series converges absolutely for \(\Re s = \sigma > {\kappa +1\over 2}\), and there is analytic continuation to the whole complex plane. As for the prototype case of the Riemann zeta-function, the investigation of the distribution of values of \(\phi(s,F)\) near its critical line \(\sigma = {\kappa\over 2}\) necessitates finding asymptotics of corresponding Dirichlet polynomials, which is the aim of this paper. For an arbitrary non-zero complex number \(\omega\), one can define \(\phi^{\omega}(s,F)\) using its Euler product expansion. Writing \(\phi^{\omega}(s,F) = \sum_{m=1}^{\infty}{g_{\omega}(m)\over m^s }\), one can explicitly express the values of the multiplicative function \(g_{\omega}\) at the prime powers in terms of the Gamma function and the coefficients in the Euler product. The authors give an evaluation of the sum \(\sum_{m\leq x}g_{\omega}^{2}(m)m^{1-\kappa}\) for \(| \omega| \leq 1/2\) and \(\Re \omega^{2}>0\). From this they deduce for \(\sigma_{T}= {\kappa\over 2}+{1\over \log T}\) and \(\omega_{T} = u(2\log\log T)^{-{1\over 2}} \) that \[ \sum_{m\leq T}{g_{\omega_{T}}^{2}(m)\over m^{2\sigma_{T}}} = e^{{u^{2}\over 2}}(1+o(1)) \] uniformly in \(u,\, 0<| u| \leq u_{0}\), as \(T\to \infty\). The authors also note that their results are valid if the Fourier coefficients are replaced by their twists \(c(m)\chi(m)\) with real characters \(\chi\).
scientific article; zbMATH DE number 2046450
scientific article