March 9, 2017
Mathematics Colloquium Lecture March 9
Mike Mossinghoff, Davidson College, will present "The Liouville Function and the Riemann Hypothesis" as part of the Mathematics Department Colloquium Lecture series at 2:30 p.m. Thursday, March 9, in 122 Cardwell Hall.
The abstract for the lecture is: The Riemann hypothesis is connected to a number of questions in number theory regarding the behavior of certain arithmetic functions, including the well-known Möbius function, and its close cousin, the Liouville function, defined as the completely multiplicative arithmetic function satisfying λ(p)=−1 for each prime p. For example, Pòlya investigated the summatory function L(x)=∑n≤xλ(n), and showed that the Riemann hypothesis would follow if L(x) never changed sign for large x. We describe some connections between oscillations in this function and some of its weighted relatives with the Riemann hypothesis and other problems, including the question of the existence of linear relations among the ordinates of the zeros of the zeta function on the critical line, and we report on some recent work on these topics. This is joint work with Tim Trudgian.