We considered the canonical gravitational partition function Z associated to classical Boltzmann–Gibbs (BG) distribution e??HZ. It is popularly thought that it cannot be built up because integral involved in constructing diverges at origin. Contrariwise, was shown (Physica A 497 (2018) 310), by appeal sophisticated mathematics developed second half of last century, this not so. can indeed compu...

The spectral correlations of the QCD Dirac operator in the 4-dimensional Euclidean box, 1/ΛQCD ≪ L ≪ 1/mπ, usually denoted as Leutwyler-Smilga regime [3], have been the object of intense study in recent years. They contain valuable information about the chiral properties of the QCD vacuum. Provided that chiral symmetry is spontaneously broken, the above regime can be described either by an effe...

Let D(s) = ∑∞ m=1 amm −s be the Dirichlet series generated by the infinite product ∏∞ k=2(1− k−s). The value of am for squarefree integers m with n prime factors depends only on the number n, and we let f(n) denote this value. We prove an asymptotic estimate for f(n) which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma.

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus t whe...