ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
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The main topic of this dissertation is the Lagarias-Odlyzko analytic algorithm for computing π(x)—the number of primes up to x. This algorithm is asymptotically the fastest known algorithm for the computation of π(x). It uses numerical integration of a function related to the Riemann zeta function, in combination with a summation involving a “kernel” function evaluated at prime powers near x. Our work resolves many issues left untreated in the original paper by Lagarias and Odlyzko and makes several original contributions—some of which have applications in other areas. In this dissertation we: (1) introduce a kernel function which appears to be more effective than the one suggested by Lagarias and Odlyzko; (2) perform a careful analysis of various sources of truncation error; (3) give choices of parameters which bound the truncation error while keeping computation to a minimum; (4) develop two new methods for enumerating primes in intervals, which require much less memory than previously known sieving methods and which are much faster than methods which test primality of single numbers; (5) describe a new method for computing ζ(s) which gives more accurate values with less complexity than classical methods.