Large sieve inequalities with power moduli and Waring’s problem

We improve the large sieve inequality with k t h kᵗʰ -power moduli, for all k ≥ 4 k≥ 4 . Our method relates these inequalities to a variant of Waring’s problem with restricted k t h kᵗʰ -powers. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two k t h kᵗʰ -powers. Secondly, we apply Marmon’s bound on the number of representations of a positive integer as a sum of four k t h kᵗʰ -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+1 k t h kᵗʰ -powers.