  The Riemann zeta function on the critical line can be computed using a
straightforward application of the Riemann-Siegel formula, Sch\"onhage's
method, or Heath-Brown's method. The complexities of these methods have
exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new
fast and potentially practical methods to compute zeta are presented. One
method is very simple. Its complexity has exponent 2/5. A second method relies
on this author's algorithm to compute quadratic exponential sums. Its
complexity has exponent 1/3. The third method employs an algorithm, developed
in this paper, to compute cubic exponential sums. Its complexity has exponent
4/13 (approximately, 0.307).
