Multiplicative Number Theory and Irregularities of Distribution

9800653 Baker该奖项为以下六个问题的调查提供资金。(1)几乎素数变量的拉格朗日四平方定理；PI的目的是通过使用质因数少得多的数字来提高布鲁登和福里的结果。（ii）顶点形式的Kloosterman和和傅立叶系数；PI将重新审视J. M. Deshouillers和H. Iwaniec的这一经典著作，以期锐化Kloosterman和和的适用界限。（iii） Bartan-Davenport-Halberstam型和的下界。在等差数列中素数的数量与期望值之间的差异的“方差”已知在某些范围内至少是预期的数量级。PI的目标是扩大这些范围。（iv）短间隔内的几乎素数：许多学者研究过的一个筛子问题，在这个问题中，一个数最多有两个素数因子，必须证明它在短间隔内存在。(v)一组点相对于凸体的差异。这里取具有光滑边界的凸集的扩展平移中的n点集的“期望”数与实际成员数之间的差值；Drmota具有明显的下界，PI旨在削弱其平滑条件。（vi）点集相对于球的差值：取上述一个点集在一个球上的期望点数与实际点数的差值。PI试图强化霍尔特版本的Erdos-Turan定理，该定理将指数和的加权平均值与发生的最大差异联系起来。（其中一些问题将由格林·哈曼（Glyn Harman）共同解决。）本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，主要研究质数、完全平方和等问题。它是数学中最古老的分支之一，在17世纪的费马手中享受着创造力的激增，并从那时起不断成长和发展。在过去的半个世纪里，它已经应用于数据传输和处理以及通信系统中的重要问题。

9800653 Baker This award provides funds for an investigation into the following six problems. (i) Lagrange's four squares theorem with almost prime variables; the PI aims to sharpen the results of Brudern and Fouvry by using numbers with far fewer prime factors. (ii) Kloosterman sums and Fourier coefficients of cusp forms; the PI shall reexamine this classic work of J. M. Deshouillers and H. Iwaniec with a view to sharpening the applicable bounds on sums of Kloosterman sums. (iii) Lower bounds for sums of Bartan-Davenport-Halberstam type. A 'variance' of differences between the number of primes in arithmetic progressions and the expected value is known to be at least of the expected order of magnitude in certain ranges. The PI aims to extend these ranges. (iv) Almost primes in short intervals: a sieve problem on which many scholars have worked, where a number with at most two prime factors must be shown to exist in a short interval. (v) Discrepancy of a set of points with respect to a convex body. Here the difference is taken between the 'expected' number and the actual number of members of an N-point set in a dilated translation of a convex set with smooth boundary; Drmota has sharp lower bounds and the PI aims to weaken his smoothness conditions. (vi) Discrepancy of point sets with respect to balls: the difference is taken between expected number and actual number of points of a set as above in a ball. The PI seeks to sharpen Holt's version of the Erdos-Turan theorem, which relates the largest difference that occurs to a weighted average of exponential sums. (Work on some of these problems will be joint with Glyn Harman.) This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with prime numbers, sums of perfect squares, and so on. It is among the oldest branches of mathematics, enjoying a surge of creativity at the hands of Fermat in the 17th century and having grown and developed continuously since then. Within the last half century it has become applicable to important problems in data transmission and processing, and communications systems.