Consider series of the form $S=\sum_{n\ge1}f(n)P(n)$, where $f$ is some smooth function, and $P$ is a periodic or quasi-periodic function (e.g., $P$ can be a trigonometric function, so $S$ a Fourier series). I would like to compute $S$ numerically to a reasonably large number of decimals (say 50). If $P$ is periodic and smooth, I usually know how to do this. But otherwise, I run into problems. Consider the following two specific examples:

1) $$S_k(a)=\sum_{n\ge1}\dfrac{J_0(na)}{n^k}\;,$$ with $J_0$ a $J$-Bessel function. If $a$ is a rational multiple of $\pi$ the sum is very regular and I can compute it. Assume otherwise. When $k\ge1$, using the integral representation of $J_0$ I can compute the sum $S_k(a)$. But when $k=0$ I have no idea, although I believe that the series converges: so question 1:

Does the series $\sum_{n\ge0}J_0(n)$ converge, and if yes, how to compute 50 decimals of its sum ?

2) $$T_k(a)=\sum_{n\ge1}\dfrac{\{na\}-1/2}{n^k}\;.$$ This series was considered by Hecke. How to compute it numerically when $a$ is irrational ? Since Hecke gives some formulas when $a$ is quadratic, specific question 2:

is it possible to compute numerically to 50 decimals $\sum_{n\ge1}\{na\}/n^2$ for $a=\pi$, and if not, for $a=\sqrt{2}$ for instance ?

  • 2
    $\begingroup$ $S_0(a)$ can be summed using Poisson summation formula and the fact that $J_0(x)$ is band limited, see en.wikipedia.org/wiki/Schl%C3%B6milch%27s_Series $\endgroup$ – user82588 Feb 5 at 12:15
  • $\begingroup$ Thanks, this completely answers my first question. $\endgroup$ – Henri Cohen Feb 5 at 17:28
  • $\begingroup$ For the second question, there might be some hope in the case $a=\sqrt{2}$ using the (very explicit) formulas for the continued fraction expansion of $\sqrt{2}$. The case $a=\pi$ seems to be much harder in this regard. $\endgroup$ – Kurisuto Asutora Feb 6 at 21:19

R.B. Paris, An expansion for the sum of a product of an exponential and a Bessel function (2019), equation (2.2): $$\sum_{n=1}^\infty J_0(na)=\frac{1}{a}-\frac{1}{2},\;\;a>0.$$ see also The evaluation of single Bessel function sums (2018).
Nemo pointed out this should further be restricted to $0<a<2\pi$.

| cite | improve this answer | |
  • $\begingroup$ Your result is wrong, see en.wikipedia.org/wiki/Schl%C3%B6milch%27s_Series . This is only valid for $0<a<2\pi$. $\endgroup$ – user82588 Feb 5 at 14:12
  • $\begingroup$ thanks, at least it answers the question in the OP: "Does the series $\sum_{n=1}^\infty J_0(n)$ converge, and if yes, how to compute 50 decimals of its sum?" The answer being $1/2$. $\endgroup$ – Carlo Beenakker Feb 5 at 14:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.