Analytic Computation of the Prime-counting Function
University of Illinois at Urbana-Champaign, 2004 - 346 pages
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 zeta(s) which gives more accurate values with less complexity than classical methods.
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