Multiple recurrence and convergence along the primes

Let $E\subset \Bbb{Z}$ be a set of positive upper density. Suppose that $P_1,P_2,\ldots,P_k\in {\Bbb Z}[X]$ are polynomials having zero constant terms. We show that the set $E\cap (E-P_1(p-1))\cap\cdots\cap (E-P_k(p-1))$ is non-empty for some prime number $p$. Furthermore, we prove convergence in $L^2$ of polynomial multiple averages along the primes.

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