Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an average along 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 $ is the von Mangoldt function and $\phi $ is the totient function. We establish improving and maximal inequalities for these averages. These bounds are uniform in the choice of progression. For instance, for $ 1< r < \infty $ there is an integer $N _{ y, r}$ so that \begin{ align*} \lVert \sup _{ N>N _{ y,r}} \lvert A_{ N,y,b} f \rvert \rVert_{ r}\ll \lVert f\rVert_{ r}. \end{ align*} The implied constant is only a function of $ r$. The uniformity over progressions imposes several novel elements on the proof.
Categories: math.CA math.NT