Higher order Laguerre inequalities for the partition function

Abstract

The Laguerre inequality and its higher order generalizations have been proved to be closely relative with the Laguerre-Pólya class and Riemann hypothesis. Recently Wagner proved the partition function satisfies the Laguerre inequality of any order as n→∞ and conjectured the thresholds of the Laguerre inequality of order m for 3≤m≤10. In this paper, for 3≤m≤10, we will give an integer N(m) such that the partition function satisfies the Laguerre inequality of order m for n>N(m).