Research Problems in Function Theory — Problem 2.45

Definitions and mathematical statement

The Bessel function of the first kind and order zero is the entire even function \[ J_0(z)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(k!)^2}\left(\frac z2\right)^{2k}. \] For each angle $\theta\in[0,2\pi)$ consider the open ray $R_\theta=\{te^{i\theta}:t>0\}$. Is it true that \[ \#\{t>0:J_0(te^{i\theta})=1\}\le1 \qquad\text{for every }\theta? \] The count is of distinct nonzero solutions, not their analytic multiplicities. The common vertex is excluded because $J_0(0)=1$ automatically; including it would change the intended ray question. Equivalently, can two distinct nonzero roots of $J_0(z)-1$ ever have the same argument modulo $2\pi$? The evenness of $J_0$ pairs roots on opposite rays and does not violate the proposed assertion. The question asks for a global statement covering all arguments and all moduli. Large-argument asymptotic estimates alone, even when they bound the number of roots on a fixed ray, do not decide uniqueness among the remaining finite roots. The source connects this assertion to removing an exceptional set in a theorem of Delsarte and Lions.

Short English statement

Can the equation $J_0(z)=1$ have two distinct nonzero solutions on the same ray from the origin?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302045 (AMR-022-2045), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.45. Original source indexed by AMR-022-2045. https://arxiv.org/abs/1809.07200.