Structure and Randomness in the Prime Numbers

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Abstract

We give a quick tour through some topics in analytic prime number theory, focusing in particular on the strange mixture of order and chaos in the primes. For instance, while primes do obey some obvious patterns (e.g. they are almost all odd), and have a very regular asymptotic distribution (the prime number theorem), we still do not know a deterministic formula to quickly generate large numbers guaranteed to be prime, or to count even very simple patterns in the primes, such as twin primes p,p+2. Nevertheless, it is still possible in some cases to understand enough of the structure and randomness of the primes to obtain some quite nontrivial results.