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Abstract: There is a beautiful connection between the prime numbers and the zeros of Riemann zeta-function in the complex plane, and understanding the distribution of these zeros can lead to remarkable consequences in our understanding of the primes. A famous open problem, known as the Riemann Hypothesis, states that the non-real zeros of the Riemann zeta-function lie on a vertical line. I will motivate this conjecture, describe some evidence for it, and describe some of the consequences it would lead to in our understanding of the primes. In particular, I will describe how to bound on the maximum gap between prime numbers assuming the Riemann Hypothesis (which is based on joint work with Carneiro and Soundararajan).