Number Theory Seminar
Wednesday, November 8, 2017, 3:15pm, 119 Physics
Michael Mossinghoff (Davidson College)
Oscillation problems in number theory
Abstract:- 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) = Σn≤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. [video]
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