Elbert–Laforgia conjecture on McMahon bounds for Bessel zeros

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Let jν,κj_{\nu,\kappa} denote the κ\kappath positive zero of the Bessel function JνJ_\nu, and let McMahon's expansion be the asymptotic expansion for these zeros. For 12<ν<12-\frac{1}{2}<\nu<\frac{1}{2}, an even (respectively odd) number of terms of McMahon's expansion always gives an upper (respectively lower) bound for jν,κj_{\nu,\kappa}. Elbert–Laforgia conjecture. For 12<ν<12-\frac{1}{2}<\nu<\frac{1}{2}, an even (respectively odd) number of terms of McMahon's expansion always gives an upper (respectively lower) bound for jν,κj_{\nu,\kappa}. This conjecture extends known bounds for positive zeros of Bessel functions and concerns the sign and usefulness of successive truncations of McMahon's asymptotic expansion. The supplied text does not state whether it has been resolved.

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Primary source

Gergő Nemes, “Proofs of two conjectures on the real zeros of the cylinder and Airy functions”, arXiv:2010.13069 (2021).

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