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arXiv · 2303.07613

Large sieve inequalities with power moduli and Waring's problem

Stephan Baier·Sean B. Lynch
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Abstract

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k$th-powers. Secondly, we input a recent and general result of Wooley on mean values of exponential sums. 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 $k+1$ $k$th-powers.

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Stephan Baier, Sean B. Lynch. 2023-03-14. Large sieve inequalities with power moduli and Waring's problem. https://doi.org/10.1090/proc/16947

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