Analytic Number Theory is a branch of mathematics that uses techniques from analysis to solve problems about integers, particularly prime numbers. It focuses on understanding the distribution of primes and their properties through tools like complex analysis and Fourier analysis. One of the key achievements in this field is the Prime Number Theorem, which describes how the number of primes less than a given number approximates the logarithmic function. Researchers in analytic number theory often explore questions related to divisor functions and zeta functions, contributing to deeper insights into the nature of numbers.