Sign changes in sums of the Liouville function

Abstract

The Liouville function $\lambda (n)$ is the completely multiplicative function whose value is $-1$ 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 $n$ 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 $L(n)=\sum _{k=1}^n\lambda (k)$, and we determine some new local extrema for this function, including some new positive values.