  In this paper we establish a general asymptotic formula for the sum of the
first $n$ prime numbers, which leads to a generalization of the most accurate
asymptotic formula given by Massias and Robin. Further we prove a series of
results concerning Mandl's inequality on the sum of the first $n$ prime
numbers. We use these results to find new explicit estimates for the sum of the
first $n$ prime numbers, which improve the currently best known estimates.
