On the Functional Independence of Zeta-Functions of Certain Cusp Forms

The zeta-function ζ ( s, F), s = σ + it of a cusp form F of weight κ in the half-plane σ > (κ + 1) / 2 is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form F. The compositions V(ζ(s,F)) with an operator V on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators V is proved.