Numerical calculation of Bessel, Hankel and Airy functions

The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument is to be evaluated. The coefficients in the well-known uniform asymptotic expansions have a complex mathematical structure which involves Airy functions. For Bessel and Hankel functions, we present an adapted algorithm which relies on a combination of three methods: (i) numerical evaluation of Debye polynomials, (ii) calculation of Airy functions with special emphasis on their Stokes lines, and (iii) resummation of the entire uniform asymptotic expansion of the Bessel and Hankel functions by nonlinear sequence transformations. In general, for an evaluation of a special function, we advocate the use of nonlinear sequence transformations in order to bridge the gap between the asymptotic expansion for large argument and the Taylor expansion for small argument ("principle of asymptotic overlap"). This general principle needs to be strongly adapted to the current case, taking into account the complex phase of the argument. Combining the indicated techniques, we observe that it possible to extend the range of applicability of existing algorithms. Numerical examples and reference values are given.Comment: 18 pages; 7 figures; RevTe

A Ces\`aro Average of Hardy-Littlewood numbers

Let $\Lambda$ be the von Mangoldt function and $r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} \Lambda(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\begin{align}\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)} &= \frac{\pi^{1 / 2}}2 \frac{N^{3 / 2}}{\Gamma(k + 5 / 2)} - \frac 12 \frac{N}{\Gamma(k + 2)} - \frac{\pi^{1 / 2}}2 \sum_{\rho} \frac{\Gamma(\rho)}{\Gamma(k + 3 / 2 + \rho)} N^{1 / 2 + \rho}\\ &+ 1/2 \sum_{\rho} \frac{\Gamma(\rho)}{\Gamma(k + 1 + \rho)} N^{\rho} + \frac{N^{3 / 4 - k / 2}}{\pi^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 \pi \ell N^{1 / 2})}{\ell^{k + 3 / 2}}\\ &- \frac{N^{1 / 4 - k / 2}}{\pi^k} \sum_{\rho} \Gamma(\rho) \frac{N^{\rho / 2}}{\pi^\rho} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2 + \rho} (2 \pi \ell N^{1 / 2})} {\ell^{k + 1 / 2 + \rho}} + \mathcal{O}_k(1).\end{align}$$ for $k > 1$, where $\rho$ runs over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$ and $J_{\nu} (u)$ denotes the Bessel function of complex order $\nu$ and real argument $u$.Comment: submitte