Michael Mossinghoff, Davidson College
“Oscillations in sums involving the Liouville function”
The Liouville function λ(n) is the completely multiplicative arithmetic function defined by λ(p) = −1 for each prime p. Po ́lya investigated its summatory function L(x) = n≤x λ(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, oscillations in L(x) and related functions remain of interest in analytic number theory. We describe some connections between oscillations in this function and its weighted relatives with the Riemann hypothesis and other problems in number theory, and report on some recent work on this topic. In particular, we describe a method involving substantial computation that establishes new bounds on the magnitude of the oscillations of L(x). This is joint work with T. Trudgian.
M3 3103