The Distribution of Primes: What we know, and how

This survey, intended for non-specialists, will describe our current knowledge of the distribution of prime numbers in various contexts (such as arithmetic progressions, values of polynomials, short intervals, and twin primes) and will compare it to the conjectured truth. Despite valiant efforts by over a century's worth of analytic number theorists, many of these conjectures still seem far out of reach. As we mention each conjecture and its related partial results, we will also describe how tools from other disciplines (such as complex function theory, combinatorics, and harmonic analysis) can be brought to bear on these number-theoretic problems.