We show that Artin's constant and two related number-theoretic quantities can be computed to t bits of precision using t^{3 + o(1)} bit operations. The factor implied by the symbol o(1) depends on the cost of the underlying arithmetic, but for practical purposes can be taken as log t. As a by-product of this work, we estimate the complexity of computing Bernoulli numbers and evaluating the Riemann zeta function at positive integers. We also give examples of constants that seem hard to compute, such as Brun's twin prime constant and the exact density of primes for which a given base is a primitive root. This last cannot be computed quickly unless factorization of certain RSA moduli is easy.