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

Mathematical notes. 2017. Vol. 101. No. 3. P. 488-496.

Abstract—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 Erdo˝ s formula obtained by using Ramanujan’s formula for the
number of variants of the partition of an integer into summands. TheHartley 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.