  The purpose of this work is to investigate the formation and evaporation of
the primordial black holes in the inflationary scenarios. Thermodynamic
parameters such as mass, temperature and entropy are expressed in terms of
NANOGrav frequency. By numerical calculations we show that the constraint on
the mass range $10^{-5}kg-10^{50}kg$ is well confirmed. We discuss the relation
between the redshift and the probability for gravitational wave source
populations. A new parameter associated with the frequency and Hubble rate is
presented, by which for the spectral index $n_{s}\approx 0.996$ and the Hubble
constant $H_{0}\approx 67.27km.s^{-1}.Mpc^{-1}$, the effective Hubble constant
is calculated to be $H_{eff,0}\approx 73.24km.s^{-1}.Mpc^{-1} $ which is
compatible with the observational data. We make a comparison between the Hubble
tension and the primordial perturbations and the expression of the mass loss
rate, chemical potential and central charge needed to describe the Hawking
evaporation will be established.
