  Sum rules constraining the R-current spectral densities are derived
holographically for the case of D3-branes, M2-branes and M5-branes all at
finite chemical potentials. In each of the cases the sum rule relates a certain
integral of the spectral density over the frequency to terms which depend both
on long distance physics, hydrodynamics and short distance physics of the
theory. The terms which which depend on the short distance physics result from
the presence of certain chiral primaries in the OPE of two R-currents which are
turned on at finite chemical potential. Since these sum rules contain
information of the OPE they provide an alternate method to obtain the structure
constants of the two R-currents and the chiral primary. As a consistency check
we show that the 3 point function derived from the sum rule precisely matches
with that obtained using Witten diagrams.
