A Generalization of a Classical Number-Theoretic Problem, Condensate of Zeros, and Phase Transition to an Amorphous Solid

Regularization of the Bose–Einstein distribution using a parastatistical correction, i.e., by means of the Gentile statistics, is carried out. It is shown that the regularization result asymptotically coincides with the Erdős formula obtained by using Ramanujan's formula for the number of variants of the partition of an integer into summands. The Hartley entropy regarded as the logarithm of the number of variants defined by Ramanujan's exact formula asymptotically coincides with the polylogarithm associated with the entropy of the Bose–Einstein distribution. The fact that these formulas coincide makes it possible to extend the entropy to the domain of the Fermi–Dirac distribution with minus sign. Further, the formulas for the distribution are extended to fractional dimension and also to dimension $1$, which corresponds to the Waring problem. The relationship between the resulting formulas and the liquid corresponding to the case of nonpolar molecules is described and the law of phase transition of liquid to an amorphous solid under negative pressure is discussed. Also the connection of the resulting formulas with the gold reserve in economics is considered.