  Let $ \Lambda $ denote von Mangoldt's function, and consider the averages
\begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)\Lambda(n) .
\end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and
sparse bounds for the maximal function. The simplest inequality is that for
sets $ F, G\subset [0,N]$ there holds \begin{equation*} N ^{-1} \langle A_N
\mathbf 1_{F} , \mathbf 1_{G} \rangle \ll \frac{\lvert F\rvert \cdot \lvert
G\rvert} { N ^2 } \Bigl( \operatorname {Log} \frac{\lvert F\rvert \cdot \lvert
G\rvert} { N ^2 } \Bigr) ^{t}, \end{equation*} where $ t=2$, or assuming the
Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is
proved for the maximal function $ \sup_N A_N \mathbf 1_{F}$. The inequalities
for $ t=1$ are sharp. The proof depends upon the Circle Method, and an
interpolation argument of Bourgain.
