Abstract
In number theory, the Ramanujan zeta function is a classical object and its special values bring together Fourier coefficients, modular transformations, elliptic integrals, and arithmetic periods. In this paper, we give a clear and rigorous way to study these values and to compute them with guaranteed error bounds. We begin with the discriminant modular form, use its Mellin transform, and then apply the classical elliptic modular parameter to obtain explicit one-dimensional period integrals in the critical range. We also give an exact method for generating the Ramanujan tau coefficients and a certified interval algorithm for numerical computation. The main results are a recurrence for the tau coefficients, a practical tail estimate for the Dirichlet series, and a precision theorem that gives the number of coefficients needed for any prescribed decimal accuracy. The numerical examples show that the method is stable and effective. The main advantage of the approach is that the symbolic formulas and the numerical approximations are kept separate, so each step can be checked directly and rigorously.
