2023/12/22 by D. R. Heath‐Brown, Heath-Brown, D. R.
Mathematics · #11L15 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2312.14531
openalex publication_date 2023/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that ∑n≤ N e(αn4)≪\ep,αN5/6+\ep for any \ep>0 and any quadratic irrational α∈\R-\Q. Classically one would have had the exponent 7/8+\ep for such α. In contrast to the author's earlier work \citecubweyl on cubic Weyl sums (which was conditional on the abc-conjecture), we show that the van der Corput AB-steps are sufficient for the quartic case, rather than the BAAB-process needed for the cubic sum.