论文标题
短时间间隔II
Multiplicative functions in short intervals II
论文作者
论文摘要
我们确定在几乎所有短时间间隔中以质数的正比例消失的乘法函数的行为。此外，我们用均匀的功率上限量化“几乎所有”，也就是说，无论间隔多长时间或短时间，我们都可以节省适当归一化的间隔长度的功率。即使在Möbius函数的特殊情况下，这种节能范围也是新的。这些总体结果是由多个应用程序激励的。首先，我们通过建立一个在几乎所有短时间间隔的整数中建立一个是两个正方形的整数数量来加强Hooley对两个正方形的总和。以前仅知道数量级。其次，我们将此结果扩展到任意数字字段$ k $的一般规范形式（两个正方形的总和是$ \ mathbb {q}（i）$的标准形式）。第三，Hooley确定了$（s_ {n + 1} - s_ {n}）^γ$的总和，$γ\ in（1，5/3）$中，其中$ s_ {1} <s_2 <s_2 <\ ldots $ deote $ deote Integers代表了两个Squares的总和。我们以$γ\在（1、3/2）$和$ s_n $中的$γ\建立了类似的结果。整数的顺序可作为任意数字字段$ k $的标准形式。这是一个大于两个的学位字段的第一个结果。假设所有Hecke $ l $ functions的Riemann假设我们还表明（1,2）$ in（1,2）$是可以接受的。第四，我们在希思棕色的最新结果中提高了关于$ x^{\ varepsilon} $平滑数字之间差距的差距。更普遍地，我们获得有关乘法序列之间差距的结果。最后，我们的结果在其他情况下也很有用，例如，在即将上映的傅立叶统一工作（与Terence Tao，JoniTeraväinen和Tamar Ziegler的联合）中。
We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power of the suitably normalized length of the interval regardless of how long or short the interval is. Such power-saving bounds are new even in the special case of the Möbius function. These general results are motivated by several applications. First, we strengthen work of Hooley on sums of two squares by establishing an asymptotic for the number of integers that are sums of two squares in almost all short intervals. Previously only the order of magnitude was known. Secondly, we extend this result to general norm forms of an arbitrary number field $K$ (sums of two squares are norm-forms of $\mathbb{Q}(i)$). Thirdly, Hooley determined the order of magnitude of the sum of $(s_{n + 1} - s_{n})^γ$ with $γ\in (1, 5/3)$ where $s_{1} < s_2 < \ldots$ denote integers representable as sums of two squares. We establish a similar results with $γ\in (1, 3/2)$ and $s_n$ the sequence of integers representable as norm-forms of an arbitrary number field $K$. This is the first such result for a number field of degree greater than two. Assuming the Riemann Hypothesis for all Hecke $L$-functions we also show that $γ\in (1,2)$ is admissible. Fourthly, we improve on a recent result of Heath-Brown about gaps between $x^{\varepsilon}$-smooth numbers. More generally, we obtain results about gaps between multiplicative sequences. Finally our result is useful in other contexts aswell, for instance in our forthcoming work on Fourier uniformity (joint with Terence Tao, Joni Teraväinen and Tamar Ziegler).
