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Non-relativistic field theories with anisotropic scale invariance in (1+1)-dare typically characterized by a dispersion relation $E\sim k^{z}$ anddynamical exponent $z>1$. The asymptotic growth of the number of states ofthese theories can be described by an extension of Cardy formula that dependson $z$. We show that this result can be recovered by counting the partitions ofan integer into $z$-th powers, as proposed by Hardy and Ramanujan a centuryago. This gives a novel relationship between the characteristic energy of thedispersion relation with the cylinder radius and the ground state energy. Forfree bosons with Lifshitz scaling, this relationship is shown to be identicallyfulfilled by virtue of the reflection property of the Riemann $\zeta$-function.The quantum Benjamin-Ono$_{2}$ (BO$_{2}$) integrable system, relevant in theAGT correspondence, is also analyzed. As a holographic realization, we providea special set of boundary conditions for which the reduced phase space ofEinstein gravity with a couple of $U(1)$ fields on AdS$_3$ is described by theBO$_{2}$ equations. This suggests that the phase space can be quantized interms of quantum BO$_{2}$ states. Indeed, in the semiclassical limit, theground state energy of BO$_{2}$ coincides with the energy of global AdS$_{3}$,and the Bekenstein-Hawking entropy for BTZ black holes is recovered from theanisotropic extension of Cardy formula.

Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy