# Michael Mossinghoff : Oscillation problems in number theory

The Liouville function λ(*n*) is the completely
multiplicative arithmetic function defined by λ(*p*) =
−1 for each prime *p*. Pólya investigated its summatory
function *L*(*x*) = Σ_{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 due
to their connections to the Riemann hypothesis and other questions in
number theory. We describe some connections between the zeta function and a
number of oscillation problems, including Pólya's question and some
of its weighted relatives, and, in joint work with T. Trudgian,
describe a method involving substantial computation that establishes new
lower bounds on the size of these oscillations.

**Category**: Number Theory**Duration**: 01:34:38**Date**: November 8, 2017 at 3:10 PM**Views**: 165-
**Tags:**seminar, Number Theory Seminar

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