LARGE SIEVE INEQUALITIES WITH POWER MODULI AND WARING'S PROBLEM

. We improve the large sieve inequality with kth-power moduli, for all k >= 4. Our method relates these inequalities to a variant of Waring's problem with restricted kth-powers. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two kthpowers. Secondly, we apply Marmon's bound on the number of representations of a positive integer as a sum of four 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 kthpowers.