Dynamical Methods for Rapid Computations of L-Functions

The primary focus of this thesis is using dynamical ideas to rapidly compute L-functions.

The main results can be summarized as: Rapid algorithm in the T-aspect.

Let Gamma be a lattice of SL(2, R ) and let f be a holomorphic or Maass cusp form on Gamma\ H .

We use the slow divergence of the horocycle flow in Gamma\SL(2, R ) to get an algorithm to compute L(f, 1/2+iT) up to a maximum error O( T-gamma) using O( T7/8+eta) operations.

Here gamma and eta are any positive numbers and the constants in O are independent of T.

We hence improve the current approximate functional equation based algorithms which have complexity O( T1+eta).

Rapid algorithm in the q-aspect. Let Gamma = SL(2, Z ), f a modular cusp form on Gamma\ H and chiq be a Dirichlet character on Z /q Z .

Let q = MN. Here M = M1, M2 such that M1|N and (M2, N) = 1, where q, M, N, M1, M 2 are integers.

We use the dynamics of the Hecke orbits to get an algorithm to compute L(f x chi q, 1/2) up to any given error O(q -gamma) using O( M5 + N) operations.

In the case when q has a factor less than q1/5, we improve current approximate functional equation based algorithms which need O(q) time complexity.

Our algorithm is most effective when q has a suitable factor of size q1/6.