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Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an averagealong the primes in a progression of diameter $ y$, given by integer $ (b,y)=1$. \begin{align*} A_{N,y,b} := \frac{\phi (y)}{N} \sum _{\substack{n<N\\n\equiv b\pmod{y}}} \Lambda (n) f(x-n) \end{align*} Above, $\Lambda $ isthe von Mangoldt function and $\phi $ is the totient function. We establishimproving and maximal inequalities for these averages. These bounds are uniformin the choice of progression. For instance, for $ 1< r < \infty $ there is aninteger $N _{y, r}$ so that \begin{align*} \lVert \sup _{N>N _{y,r}} \lvertA_{N,y,b} f \rvert \rVert_{r}\ll \lVert f\rVert_{r}. \end{align*} The impliedconstant is only a function of $ r$. The uniformity over progressions imposesseveral novel elements on the proof.

Improving and Maximal Inequalities for Primes in Progressions