New analytic algorithms in number theory

1. In t roduc t ion . Several recently invented number-theoretic algorithms are sketched below. They all have the common feature that they rely on bounded precision computations of analytic functions. The main one of these algorithms is a new method of calculating values of the Riemann zeta function at multiple points. This method enables one to verify the truth of the Riemann Hypothesis (RH) for the first n zeros much faster than was possible before. Methods for the approximate computation of zeta and L-functions have also been shown to lead to fast algorithms for the exact computation of certain integer-valued numbertheoretic functions. Numerical computations of zeros of the zeta function cannot ever prove the RH, but if there is a counterexample to the RH, such computations might find it. This was the main motivation for most of the computations that have been carried out so far. The first few nontrivial zeros were computed by hand by Riemann [5, 23], and further computations, by hand or with the help of electromechanical devices, were carried out in the early 1900s, culminating with the verification of the RH for the first 1041 zeros (i.e., the 1041 zeros in the upper half-plane that are closest to the real axis) by Titchmarsh and Comrie [25]. After World War II, these computations were extended using electronic digital computers. The latest result in this direction is the verification of the RH for the first 1.5 • 10 zeros [16], a computation that involved about two months of time on a modern supercomputer, and used the same basic method for computing the zeta function that was employed by Titchmarsh and Comrie. Computations of the zeros of the Riemann zeta function and related functions (such as Dirichlet L-functions and Epstein zeta functions) are also useful in providing evidence for and against various other conjectures, such as the Montgomery pair correlation conjecture [17, 18], and some related conjectures which predict that the zeros of ç(s) ought to behave like eigenvalues of random hermitian matrices (whose distribution is known relatively well, since they have been studied by physicists who use them to model energy levels in many-particle systems). For a discussion of these conjectures and the numerical evidence about them, see [17, 18, 19, 20]. Another motivation for computing zeros of zeta