Summary: This survey is devoted to the classical and modern problems related to the entire function \({\sigma({\mathbf{u}};\lambda)} \), defined by a family of nonsingular algebraic curves of genus 2, where \({\mathbf{u}} = (u_1,u_3)\) and \(\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})\). It is an analogue of the Weierstrass sigma function \(\sigma({{u}};g_2,g_3)\) of a family of elliptic curves. Logarithmic derivatives of order \(2\) and higher of the function \({\sigma({\mathbf{u}};\lambda)}\) generate fields of hyperelliptic functions of \({\mathbf{u}} = (u_1,u_3)\) on the Jacobians of curves with a fixed parameter vector \(\lambda \). We consider three Hurwitz series \(\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}, \sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}\) and \(\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!} \). The survey is devoted to the number-theoretic properties of the functions \(a_{m,n}(\lambda), \xi_k(u_1;\lambda)\) and \(\mu_k(u_3;\lambda)\). It includes the latest results, which proofs use the fundamental fact that the function \({\sigma ({\mathbf{u}};\lambda)}\) is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.