Radii of starlikeness and convexity of a cross-product of Bessel functions

To illustrate vertical and horizontal equidistributions of Kloosterman sums, we prove two relevant central limit theorems by some analytic and combinatorial properties of Chebyshev polynomials and independence of monodromy groups of Kloosterman sheaves. The first one supports a short interval version of Katz’s vertical Sato–Tate distribution theorem, and the second one coincides with the horizontal conjecture of Katz in the statistical sense.