History of Zeros of Bessel functions of the first kind $J_\alpha$

   rigour: heuristic (agreement-checked)   rigour details: 'Computed twice at different working precisions, keeping only the-    digits both computations support. That bounds the error from working precision-    and nothing else: two runs of the same method agree even when the method is wrong.'+    digits both computations support. Since 2026-08-17 each value is also *proven+    to bracket a zero*: the Bessel function is evaluated in ball arithmetic at both+    ends of the interval the written digits denote, and the two results have strictly+    opposite signs, so a zero lies between them by the intermediate value theorem.+    All entries pass. What this does not establish is the index -- that this is the+    nth zero rather than a neighbour -- which needs a count of the zeros below it,+    and arb''s counting is not exposed here.' Display properties:   number-header: $n$<sup>th</sup> root of $J_\alpha$