Michael Mossinghoff

Affiliation: Davidson College

Email: mimossinghoff@davidson.edu

Title Of Talk: Oscillations in sums involving the Liouville function

Abstract: The Liouville function $\lambda(n)$ is the completely multiplicative arithmetic function defined by $\lambda(p) = -1$ for each prime $p$. Pólya investigated its summatory function $L(x) = \sum_{n\leq x} \lambda(n)$, and showed for instance that the Riemann hypothesis would follow if $L(x)$ never changed sign for large $x$. While it has been known since the work of Haselgrove in 1958 that $L(x)$ changes sign infinitely often, the behavior of $L(x)$ and related functions remain of interest in analytic number theory. For example, some of K.~Alladi's early work concerns some restricted sums of the closely related Möbius function. We describe some recent work that establishes new bounds on the magnitude of the oscillations of $L(x)$ and its weighted relatives, $L_\alpha(x) = \sum_{n\leq x} \lambda(n)/n^\alpha$, where $0\leq\alpha\leq1$. This is joint work with T. Trudgian.

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