Sign changes in sums of the Liouville function

The Liouville function λ(n) is the completely multiplicative function whose value is −1 at each prime. We develop some algorithms for computing the sum T (n) = ∑n k=1 λ(k)/k, and use these methods to determine the smallest positive integer n where T (n) < 0. This answers a question originating in some work of Turán, who linked the behavior of T (n) to questions about the Riemann zeta function. We also study the problem of evaluating Pólya’s sum L(n) = ∑n k=1 λ(k), and we determine some new local extrema for this function, including some new positive values.