Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions.
Historically the subject has been dominated by the prime number theorem, first by attempts to prove it and then by improvements in the error term.
The distribution of primes numbers among residue classes modulo an integer is an area of active research.
There is also much interest in the size of the smallest prime in an arithmetic progression; Linnik's theorem gives an estimate.
The large sieve and exponential sums are usually considered part of multiplicative number theory.