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Sign changes in sums of the Liouville function

Authors:	Peter Borwein Ron Ferguson Michael J Mossinghoff

Institution:	Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A 1S6 Canada ; Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A 1S6 Canada ; Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996

Abstract:	The Liouville function $\lambda(n)$ is the completely multiplicative function whose value is at each prime. We develop some algorithms for computing the sum $T(n)=\sum_{k=1}^n \lambda(k)/k$ , and use these methods to determine the smallest positive integer where . This answers a question originating in some work of Turán, who linked the behavior of to questions about the Riemann zeta function. We also study the problem of evaluating Pólya's sum $L(n)=\sum_{k=1}^n\lambda(k)$ , and we determine some new local extrema for this function, including some new positive values.

Keywords:	Liouville function P\'olya's sum Tur\'an's sum Riemann hypothesis

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