Prime Numbers in Short Intervals and a Generalized Vaughan Identity

Abstract

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.