Multiplicative Number Theory and Irregularities of Distribution

小行星9800653 该奖项为调查以下六个问题提供资金。(i)拉格朗日的四平方定理与几乎素变量; PI的目的是锐化结果Brudern和Fouvry使用的数字与少得多的素因子。(ii)Kloosterman和和尖点形式的傅立叶系数; PI将重新审查这一经典工作的J。Deshouillers和H. Iwaniec的一个推广，目的是提高Kloosterman和的适用界. (iii)Bartan-Davenport-Halbernut型和的下界 已知算术级数中素数的数量与期望值之间的差异的“方差”在某些范围内至少是期望的数量级。 PI旨在扩大这些范围。(iv)短区间内的几乎素数：许多学者研究的一个筛子问题，其中一个最多有两个素因子的数必须在短区间内存在。(v)点集关于凸体的离散性。 这里的差异是采取之间的“预期”的数量和实际数量的成员的N点集在一个扩张的翻译凸集与光滑的边界; Drmota有尖锐的下界和PI的目的是削弱他的光滑条件。 (vi)关于球的点集的离散性：在球中的期望数量和如上所述的集合的点的实际数量之间取差。 PI试图锐化霍尔特版本的埃尔多斯-图兰定理，该定理将发生的最大差异与指数和的加权平均值联系起来。(Work关于其中一些问题，将与格林·哈曼联合讨论。） 本研究属于一般数学领域的数论研究福尔斯。数论的历史根源在于对整数的研究，解决诸如素数、完全平方和等问题，它是数学中最古老的分支之一，世纪费马的创造力使其蓬勃发展，并从那时起不断发展壮大。在过去的半个世纪，它已成为适用于重要的问题，在数据传输和处理，以及通信系统。

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.