This presentation serves as an introduction to Additive Number Theory, starting with the fundamental distinction between partitions and representations. We revisit the classical theory of generating functions developed by Euler and explore how Ramanujan utilized complex analysis—specifically Cauchy's integral formula—to study the asymptotic behavior of the partition function $P(n)$.
We then transition to the Hardy-Littlewood Circle Method, explaining how it generalizes these ideas to solve major challenges like Waring's Problem ($x_1^k + \dots + x_r^k = n$).
Finally, we discuss I.M. Vinogradov's contributions, specifically his refinement using finite sums over prime numbers to approach Goldbach's Weak Conjecture, and conclude with an explicit calculation of integer representations using contour integration.