Large sieve inequalities with power moduli and Waring’s problem

Abstract:

We improve the large sieve inequality with $k^{\mathrm {th}}$-power moduli, for all $k\ge 4$. Our method relates these inequalities to a variant of Waring’s problem with restricted $k^{\mathrm {th}}$-powers. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k^{\mathrm {th}}$-powers. Secondly, we apply Marmon’s bound on the number of representations of a positive integer as a sum of four $k^{\mathrm {th}}$-powers. Thirdly, we use Wooley’s Vinogradov mean value theorem with arbitrary weights. Lastly, we state a conditional result, based on the conjectural Hardy-Littlewood formula for the number of representations of a large positive integer as a sum of $k+1$ $k^{\mathrm {th}}$-powers.