Abstract. Very extensive computations are reported which extend and, partly, check previous computations concerning the location of the complex zeros of the Riemann zeta function. The results imply the truth of the Riemann hypothesis for the first 1,500,000,001 zeros of the form a + it in the critical strip with 0 < t < 545,439,823.215, i.e., all these zeros have real part a = 1/2. Moreover, all these zeros are simple. Various tables are given with statistical data concerning the numbers and first occurrences of Gram blocks of various types; the numbers of Gram intervals containing m zeros, form = O, l, 2, 3 and 4; and the numbers of exceptions to "Rosser's rule" of various types (including some formerly unobserved types). Graphs of the function Z( t) are given near five rarely occurring exceptions to Rosser's rule, near the first Gram block of length 9, near the clpsest observed pair of zeros of the Riemann zeta function, and near the largest (positive and negative) found values of Z(t) at Gram points. Finally, a number of references are given to various number-theoretical implications.

Abstract. If N is an odd perfect number, and q \\ N, q prime, k even, 2k then it is almost immediate that N > q .We prove here that, subject to certain conditions verifiable in polynomial time, in fact N > q ' . Using this and related results, we are able to extend the computations in an earlier paper to show that N > 10300 .

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Abstract. Many problems in computational number theory require the application of some sieve. Efficient implementation of these sieves on modern computers has extended our knowledge of these problems considerably. This is illustrated by three classical problems: the Goldbach conjecture, factoring large numbers, and computing the summatory function of the Mobius function.