In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function.

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There are $2^p$ subsets of $\mathbb F_p$ . All of them are sumsets, that is sets of the form $A+B:=\{a+b: a \in A, b \in B\}$ , since $A=A+\{0\}$ . However, Green and Ruzsa showed that there are only $2^{p/3+o(p)}$ sumsets $A+A \in \mathbb F_p$ , and Noga Alon, Adrian Ubis and I show that that there are no more than $1.97^p$ sumsets $A+B$ , with $|A|,|B|>1$ . Moreover we show that there are only $2^{p/2+o(p)}$ sumsets $A+B$ with $|A|,|B| > m(p)$, if $m(p)$ goes to infinity with $p$ (though arbitrarily slowly). One can ask more precise questions; for example, how many sumsets $A+B$ are there with $|A|=k$ and $|B|=l$ ? We get bounds on this, in particular showing that, when $k,l > m(p)$ (as above), one gets $2^{p/2+o(p)}$ such sumsets if and only if $k+l = p/4 + o(p)$ . The methods used are combinatorial and analytic.

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