“BEST POSSIBLE” UPPER AND LOWER BOUNDS FOR THE ZEROS OF THE BESSEL FUNCTION Jν(x)

Let jν,k denote the k-th positive zero of the Bessel function Jν(x). In this paper, we prove that for ν > 0 and k = 1, 2, 3, . . . , ν − ak 21/3 ν < jν,k < ν − ak 21/3 ν + 3 20 ak 21/3 ν1/3 . These bounds coincide with the first few terms of the well-known asymptotic expansion jν,k ∼ ν − ak 21/3 ν + 3 20 ak 21/3 ν1/3 + · · · as ν →∞, k being fixed, where ak is the k-th negative zero of the Airy function Ai(x), and so are “best possible”.