Topics in multiplicative number theory

In this thesis we use a blend of analytic and combinatorial techniques to address a variety of problems about primes and related multiplicative structures: We introduce a modified linear sieve, and obtain new upper bound for twin primes, refining a 2004 result of Wu. This represents the largest improvement on the problem since 1986. We prove the equidistribution of primes p in arithmetic progressions to moduli of size p^{0.5313}, and consequently obtain the infinitude of shifted primes p-1 without prime factors above p^{0.2844}, refining a 1998 result of Baker and Harman. We introduce a new hybrid conjecture of the famous conjectures of Hardy--Littlewood and Chowla on correlations of the Mobius and von Mangoldt functions. We prove several `on average' results on the problem (in part jointly with J. Teravainen), extending work of Matomaki, Radziwill, and Tao. We study primitive sets. The Erdos primitive set conjecture, posed in 1986, asserts that the set of primes is maximal among all primitive sets, in a precise sense. We explore variations of this problem in multiple directions, where many open questions remain. We disprove a natural generalization of the conjecture, due to Banks and Martin in 2013, by showing the k-almost primes are minimal when k=6, in the same sense of Erdos. We also show a translated analogoue of the conjecture is false already for translates h~1.04, highlighting the subtlety of the original problem. Finally, this thesis culminates with a proof of the Erdos primitive set conjecture