[Colloquium] Experimental investigations into the zeros of Incomplete Gamma Functions and certain approximates to the Riemann Xi -function
SPEAKER | Jim Haglund
INSTITUTE | University of Pennsylvania
DATE | May 13(Mon), 2013
TIME | 14:00
PLACE | 1503
ABSTRACT | The Riemann Xi-function is a certain entire function which is real on the real line. Riemann conjectured that all of the zeros of the Xi-function are real, which is now known as the Riemann Hypothesis. Riemann expressed the Xi-function as a trigonometric integral, where the integrand is an infinite series. In this talk we discuss some experimental investigations of the speaker into the zeros of the integral obtained by truncating this integrand after N steps. Based on numerical computations, we suggest that the zeros of these integrals satisfy a simple monotonicity condition, which implies the Riemann hypothesis. We also discuss some affiliated conjectures involving the Ramanujan Xi-function, and the zeros of incomplete gamma functions.