  Monte Carlo simulations and variational calculations using a Gaussian ansatz
are applied to a model consisting of a flexible linear polyelectrolyte chain as
well as to an intrinsically stiff chain with up to 1000 charged monomers.
Addition of salt is treated implicitly through a screened Coulomb potential for
the electrostatic interactions.
  For the flexible model the electrostatic persistence length shows roughly
three regimes in its dependence on the Debye-H\"{u}ckel screening length,
$\kappa^{-1}$.As long as the salt content is low and $\kappa^{-1}$ is longer
than the end-to-end distance, the electrostatic persistence length varies only
slowly with $\kappa^{-1}$. Decreasing the screening length, a controversial
region is entered. We find that the electrostatic persistence length scales as
$sqrt{\xi_p}/\kappa$, in agreement with experiment on flexible
polyelectrolytes, where $\xi_p$ is a strength parameter measuring the
electrostatic interactions within the polyelectrolyte. For screening lengths
much shorter than the bond length, the $\kappa^{-1}$ dependence becomes
quadratic in the variational calculation. The simulations suffer from numerical
problems in this regime, but seem to give a relationship half-way between
linear and quadratic.
  A low temperature expansion only reproduces the first regime and a high
temperature expansion, which treats the electrostatic interactions as a
perturbation to a Gaussian chain, gives a quadratic dependence on the Debye
length.
  For a sufficiently stiff chain, the persistence length varies quadratically
with $\kappa^{-1}$ in agreement with earlier theories.
