This paper introduces a novel axisymmetric Helmholtz phononic crystal ( AHPC) and conducts an in-depth investigation. A numerical model was developed to derive the complete bandgap of the AHPC structure, revealing the mechanism behind the formation of the bandgap, describing the characteristics of the sound pressure distribution at the edges of the bandgap, and confirming the low-frequency bandgap through experimentation, thereby validating the accuracy of the numerical model. The effects of neck length, neck width and cavity length on the low-frequency bandgap width and bandgap edge frequency were quantitatively investigated, identifying that these three parameters are the main factors affecting the bandgap distribution. Both the neck length and cavity length are positively correlated with the bandgap width and negatively correlated with the edge frequency of the bandgap, while the neck width shows a positive correlation with both bandgap width and edge frequency. Response surface analysis based on the Box-Behnken model was performed, and the functional relationships between the three factors and the bandgap width or the lower limit of the bandgap were obtained. Using these functional expressions, the structural parameters were optimized via the interior point method. The optimization results were verified by numerical simulation, and the optimal AHPC structure with a low-frequency bandgap ranging from 298. 49 Hz to 519. 27 Hz was obtained.