Decoupling, exponential sums and the Riemann zeta function (Q2826782)

This paper uses the method of \textit{E. Bombieri} and \textit{H. Iwaniec} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 449--472 (1986; Zbl 0615.10047)] to prove that NEWLINE\[NEWLINE\zeta(\tfrac12+it)\ll(1+|t|)^{13/84+\varepsilon}NEWLINE\]NEWLINE for any \(\varepsilon>0\), thereby improving on the previous record of \textit{M. N. Huxley} [Proc. Lond. Math. Soc. (3) 66, No. 1, 1--40 (1993; Zbl 0803.11046)] in which the exponent \(13/84=0.15476\ldots\) was replaced by \(32/205=0.15609\ldots\). It is also shown that NEWLINE\[NEWLINE \left(\frac{13}{84}+\varepsilon,\quad\frac{55}{84}+\varepsilon\right)NEWLINE\]NEWLINE is an exponent pair, for any \(\varepsilon>0\).NEWLINENEWLINEThe argument uses the formulation of the Bombieri-Iwaniec method due to \textit{M. N. Huxley} [Area, lattice points, and exponential sums. Oxford: Clarenton Press (1996; Zbl 0861.11002)], in which two spacing problems arise. The present paper improves over earlier works through a better result on the ``first spacing problem'', which concerns the distribution of the vectors NEWLINE\[NEWLINE(n,n^2,n^{3/2},n^{1/2})\;\;\;(1\leq n\leq N).NEWLINE\]NEWLINE The principal technical result of the paper is the twelfth moment estimate NEWLINE\[NEWLINE\int_{[-1,1]^4}\left|\Sigma(x_1,x_2,x_3,x_4)\right|^{12}dx_1dx_2dx_3dx_4\ll N^{6+\varepsilon},NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\Sigma(x_1,x_2,x_3,x_4)=\sum_{n\leq N} e(x_1n+x_2n^2+x_3N^{1/2}n^{3/2}+x_4N^{1/2}n^{1/2}).NEWLINE\]NEWLINE This is optimal (apart from the \(\varepsilon\)) and provides a strong bound for the first spacing problem.NEWLINENEWLINETo prove the above twelfth moment bound the author first establishes a general decoupling estimate for curves in \(\mathbb{R}^d\). However quite a bit more work is needed to complete the argument.