On the $\nu$-zeros of the modified Bessel function $K_{i\nu}(x)$ of positive argument

The modified Bessel function of the second kind $K_{i\nu}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $\nu_n\sim \pi n/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimate for the bahaviour of these zeros for large $n$ by making use of the known asymptotic expansion of $K_{i\nu}(x)$ for large $\nu$. Numerical results are presented to illustrate the accuracy of the expansion obtained.