Newman's method proves the PNT by showing the integral converges, and therefore the integrand goes to zero as , which is the PNT. In general, the convergence of the improper integral does not imply that the integrand goes to zero at infinity, since it may oscillate, but since is increasing, it is easy to show in this case.
Newman's method proves the PNT by showing the integral converges and therefore the integrand goes to zero as , which is the PNT. In general the convergence of the improper integral does not imply that the integrand goes to zero at infinity, since it may oscillate, but since is increasing, it is easy to show in this case