Bose–Einstein Distribution as a Problem of Analytic Number Theory: The Case of Less than Two Degrees of Freedom

The problem of finding the number and the most likely shape of solutions of the system $\sum_{j=1}^\infty\lambda_{j}n_{j}\le M$, $\sum_{j=1}^\infty n_j=N$, where $\lambda_j,M,N>0$ and $N$ is an integer, as $M,N\to\infty$, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of $\lambda_j$ is of the order of $\lambda^{d/2}$, where $d$, the number of degrees of freedom, is less than two.