Fourier uniformity of bounded multiplicative functions in short intervals on average
Fourier uniformity of bounded multiplicative functions in short intervals on average
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DOI：
10.1007/s00222-019-00926-w
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发表时间：
2019-09
影响因子：
3.1
通讯作者：
Kaisa Matomäki;Maksym Radziwill;T. Tao
中科院分区：
文献类型：
--
作者：
Kaisa Matomäki;Maksym Radziwill;T. Tao
让 表示刘维尔函数。我们将其表示为 $$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx = o ( X H) \end{对齐}$$对于所有固定但任意小的。此前，这只是众所周知的。对于较小的值，这是该尺度上平均局部傅立叶均匀性的第一个“非平凡”情况。我们还获得了（非自命不凡的）1 有界乘法函数的类似陈述。我们通过获得范围内的抵消总和来说明结果的强度，其中 是 von Mangoldt 函数。
Letdenote the Liouville function. We show that as, $$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx = o ( X H) \end{aligned}$$for allwithfixed but arbitrarily small. Previously, this was only known for. For smaller values ofthis is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum ofover the rangesand, and whereis the von Mangoldt function.