  It has been proposed that the short period gamma-ray bursts, which occur at a
rate of $\sim 10 {\rm yr^{-1}}$, may be evaporating primordial black holes
(PBHs). Calculations of the present PBH evaporation rate have traditionally
assumed that the PBH mass function varies as $M_{{\rm BH}}^{-5/2}$. This mass
function only arises if the density perturbations from which the PBHs form have
a scale invariant power spectrum. It is now known that for a scale invariant
power spectrum, normalised to COBE on large scales, the PBH density is
completely negligible, so that this mass function is cosmologically irrelevant.
For non-scale-invariant power spectra, if all PBHs which form at given epoch
have a fixed mass then the PBH mass function is sharply peaked around that
mass, whilst if the PBH mass depends on the size of the density perturbation
from which it forms, as is expected when critical phenomena are taken into
account, then the PBH mass function will be far broader than $ M_{{\rm
BH}}^{-5/2}$. In this paper we calculate the present day PBH evaporation rate,
using constraints from the diffuse gamma-ray background, for both of these mass
functions. If the PBH mass function has significant finite width, as recent
numerical simulations suggest, then it is not possible to produce a present day
PBH evaporation rate comparable with the observed short period gamma-ray burst
rate. This could also have implications for other attempts to detect
evaporating PBHs.
