COMPUTATIONS OF THE MERTENS FUNCTION AND IMPROVED BOUNDS ON THE MERTENS CONJECTURE

摘要

The Mertens function is defined as M(x) = Sigma(n = x) mu(n), where mu(n) is the Mobius function. The Mertens conjecture states vertical bar M(x)/root x vertical bar 1 for x 1, which was proven false in 1985 by showing lim inf M(x)/root x -1.009 and lim sup M(x)/root x 1.06. The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of -1.837625 and 1.826054. In addition, M(x) was computed for all x = 10(16), recording all extrema, all zeros, and 10(8) values sampled at a regular interval. Finally, an algorithm to compute M(x) in O(x(2/3+epsilon)) time was used on all powers of two up to 2(73).