Multiplicative Number Theory
Multiplicative Number Theory
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DOI：
10.1007/978-1-4757-5927-3
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
H. Davenport
中科院分区：
文献类型：
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作者：
H. Davenport
尽管《乘法数论》的出版时间很短，但它的最初版本对研究和年轻的数学家产生了重大影响。通过对大筛子和庞比耶里定理的关联描述，达文波特教授使人们能够接触到大量的新发现。在这一刺激下，取得了如此大的进步，以至于我们目前对这些主题的理解远远超出了1966年的了解。因为现在可以更容易地证明主要结果。我做出了彻底的决定，为第二版完全重写了第23-29节。在进行这些改动时，我试图保持原作的基调和精神。我没有像达文波特那样从L时间的零密度估计推导庞比耶里定理，而是选择给出沃恩对庞比耶里定理的初等证明。这种方法依赖于Vaughan估计素数之和的Vinogradov方法的简化版本(见§24)。Vinogradov设计了他的方法来估计和LPh e(Prx)；为了保持历史观点，在转向大筛子和Bombieri定理之前，我插入了(在§25，26中)关于这个指数和及其对素数和的应用的讨论。在1969年达文波特教授英年早逝之前，几位数学家曾建议对乘法数论进行一些小的改进，如果它有一天重印的话。
Although it was in print for a short time only, the original edition of Multiplicative Number Theory had a major impact on research and on young mathematicians. By giving a connected account of the large sieve and Bombieri's theorem, Professor Davenport made accessible an important body of new discoveries. With this stimula tion, such great progress was made that our current understanding of these topics extends well beyond what was known in 1966. As the main results can now be proved much more easily. I made the radical decision to rewrite §§ 23-29 completely for the second edition. In making these alterations I have tried to preserve the tone and spirit of the original. Rather than derive Bombieri's theorem from a zero density estimate tor L timctions, as Davenport did, I have chosen to present Vaughan'S elementary proof of Bombieri's theorem. This approach depends on Vaughan's simplified version of Vinogradov's method for estimating sums over prime numbers (see § 24). Vinogradov devised his method in order to estimate the sum LPH e (prx); to maintain the historical perspective I have inserted (in §§ 25, 26) a discussion of this exponential sum and its application to sums of primes, before turning to the large sieve and Bombieri's theorem. Before Professor Davenport's untimely death in 1969, several mathematicians had suggested small improvements which might be made in Multiplicative Number Theory, should it ever be reprinted.