Abstract
We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with 1⩽s⩽k2-1. In this way, we prove the main conjecture for such even moments when 1⩽s⩽14(k+1)2, showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of s, these finding application to the asymptotic formula in Waring’s problem.
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