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

The first positive zero jv,1 of the Bessel function jv(x) has the asymptotic expansion jv,1=v−a1213v13+320 a21213v13+… where a1 = −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 jv,1, i.e., jv,1<v−a1213v13+320a21 213v13+… and that similar bounds hold for jv,k, j′v,k, yv,k and y′v,k, k = 1,2,…. The objective of this paper is to show that Lorchʹs conjecture is true when v ⩾ 10 for jv,1 and jv,2.