In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas are presented (the existence of such new formulas had been established by Lev and Olevskii).
The text introducing the paper of Meyer says :
An important problem in harmonic analysis is solved in this article: Is the Poisson summation formula unique or does it belong to a wider class? The latter is true. The method that is used to prove this statement is surprising. Our new Poisson’s formulas were hidden inside an old and almost forgotten paper published in 1959 by A. P. Guinand. The role of number theory in this issue is fascinating.
Indeed, the function $r_3$ that counts the number of decompositions of an integer into a sum of three squares makes an appearance.
While that paper of Guinand was only cited 3 times as of 1999 according to Google Scholar, one of the three is in a paper of Berndt Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications JNT Volume 7, Issue 4, November 1975, Pages 413–445 which was much more cited. Yet it appears that the type of formulas obtained by Berndt are different from those of Meyer.
Hence the question in the title :
1) do you know of any reference (published before 2016) where a Poisson-type formula involving $r_3$ appears in number theory ?
To which I'll add :
2) are these type of formulas surprising to a number theorist, or looking rather natural ?