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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 ?

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    $\begingroup$ $S_0(a)$ 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
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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 $0<a<2\pi$.

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  • $\begingroup$ 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

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