“Best possible” upper bounds for the first two positive zeros of the Bessel function Jv( x): The infinite case

Abstract

The first positive zero j v,1 of the Bessel function j v ( x) has the asymptotic expansion j v,1=v− a 1 2 1 3 v 1 3 + 3 20 a 2 1 2 1 3 v 1 3 +… where a 1 = −2.33811 is the first negative zero of the Airy function Ai( x). Recently, Lorch has conjectured that the sum of the first three terms in the expansion gives an upper bound for j v,1 , i.e., j v,1<v− a 1 2 1 3 v 1 3 + 3 20 a 2 1 2 1 3 v 1 3 +… and that similar bounds hold for j v, k , j′ v, k , y v, k and y′ v, k , k = 1,2,…. The objective of this paper is to show that Lorch's conjecture is true when v ⩾ 10 for j v,1 and j v,2 .