  We will study the asymptotic behavior of summation functions of a natural
argument, including the asymptotic behavior of summation functions of a prime
argument in the paper. A general formula is obtained for determining the
asymptotic behavior of the sums of functions of a prime argument based on the
asymptotic law of primes. We will show, that under certain conditions:
  $\sum_{p \leq n} {f(p)}= \sum_{k=2}^n {\frac {f(k)}{\log(k)}(1+o(1))}$, where
$p$ is a prime number. In the paper, the necessary and sufficient conditions
for the fulfillment of this formula are proved.
