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

AbstractThe 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