This dissertation focuses on three problems in analytic number theory, one of a multiplicative nature, and the latter two additive. The first problem concerns some generalizations of smooth numbers and Bombieri-Vinogradov-type theorems when the moduli are restricted to these numbers. In particular, we show that we can use a preliminary beta sieve to replace the multiple densely divisible numbers used in Polymath 8's \cite{Polymath8Writeup} refinement of Zhang's \cite{Zhang} theorem on bounded gaps between primes. This simplification retains the necessary asymptotic bounds, which allows us to subsequently apply the GPY sieve and preserve the bounds attained by the Polymath project.
Second, we examine the Jacobi-Faulhaber theorem and its refinements and generalizations. The original formulation of the theorem states that sums of consecutive higher powers can be written as polynomials of sums of consecutive integers and sums of consecutive squares. We generalize this result by considering what necessary and sufficient conditions on a set of exponents imply that polynomials of sums of consecutive powers of those exponents can represent sums of consecutive higher powers. We also inspect generalizations of the Jacobi-Faulhaber theorem for power-sums in arithmetic progressions, classifying the cases for which the full strength of the original theorem hold.
The third problem is a new approach to the asymptotic formula in Waring’s problem. We consider the circle method on weighted exponential sums that serve as modified generating functions in Waring's problem. Our evaluation of the contribution over the major arcs simplifies the methods used for the unweighted counterpart. By generalizing Hua's lemma, we obtain new mean value estimates for weighted exponential sums, which, when combined with our pointwise and recent decoupling results \cite{BDGVin, WooleyVin}, provides bounds that match the current bounds for the number of variables needed to represent all sufficiently large integers as sums of $k$-th powers. We also provide an equidistribution hypothesis that implies that our choice of weights is optimal.