Improving and Maximal Inequalities for Primes in Progressions

TL;DR

This paper develops uniform maximal inequalities for averages over primes in arithmetic progressions, advancing the understanding of prime distribution and inequalities in analytic number theory.

Contribution

It introduces new uniform bounds for averages over primes in progressions, improving existing inequalities and handling the uniformity across different progressions.

Findings

Established bounds for maximal functions of primes in progressions.

Proved inequalities are uniform across different arithmetic progressions.

Provided tools for further research in prime distribution and inequalities.

Abstract

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.