Analytic number theory is a branch of mathematics that uses techniques from analysis to solve problems about integers and prime numbers. It focuses on understanding the distribution of prime numbers and their properties through tools like complex analysis and Fourier analysis. One of the key achievements in analytic number theory is the Prime Number Theorem, which describes how the primes are distributed among the integers. This theorem shows that the number of primes less than a given number approximates to that number divided by the natural logarithm of that number, revealing deep connections between number theory and analysis.