Prime Numbers in Short Intervals and a Generalized Vaughan Identity
Prime Numbers in Short Intervals and a Generalized Vaughan Identity
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DOI：
10.4153/cjm-1982-095-9
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发表时间：
1982-12
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通讯作者：
D. R. Heath-Brown
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文献类型：
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作者：
D. R. Heath-Brown
1.导论.许多涉及素数的问题都依赖于对适当函数f（n）的形式为f（n）f（n）的估计和（这里，通常，Λ（n）是冯·曼戈尔特函数）。三种不同的一般方法被用来估计这些数额。最早的是Vinogradov（见[13，第9章]）;第二个涉及Dirichlet L-函数的零密度界（例如见[8，第15和16章]）;第三个是Vaughan（例如见[12]）使用了一个算术恒等式，后面会解释。第二种和第三种方法比第一种方法简单得多。另一方面，维诺格拉多夫的技术至少与沃恩的技术一样强大，有时甚至更强大。在许多情况下，沃恩恒等式比使用零密度估计产生更好的边界，但有时它们更糟。本文的目的是给出Vaughan方法的一个简单的推广，它本质上与上述任何一种方法一样强大，讨论它的一般含义，并将它应用于Huxley [4]的下列结果的证明，该结果以前只在零密度方法的范围内。
1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three distinct general methods have been used to estimate such sums. The earliest is due to Vinogradov (see [13, Chapter 9]); the second involves zerodensity bounds for Dirichlet L–functions (see [8, Chapters 15 and 16] for example); and the third, due to Vaughan (see [12] for example) uses an arithmetical identity as will be explained later. The second and third methods are much simpler to apply than the first. On the other hand Vinogradov's technique is at least as powerful as Vaughan's and occasionally more so. In many cases Vaughan's identity yields better bounds than the use of zero–density estimates, but sometimes they are worse. The object of this paper is to present a simple extension of Vaughan's method which is essentially as powerful as any of the techniques mentioned above, to discuss its general implications, and to apply it to the proof of the following result of Huxley [4], which has previously only been within the scope of the zero density method.