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Estimation of exponential sums over primes in short intervals. I. (English) Zbl 0922.11070

Let \(\Lambda(n)\) and \(\mu(n)\) be the von Mangoldt and Möbius functions, respectively. A non-trivial estimate for \(\sum_{n\leq x}\Lambda(n)e^{2\pi in\alpha}\), where \(\alpha\) is real, was first obtained by I. M. Vinogradov in his famous solution to the ternary Goldbach problem. Recently much effort has been devoted to such sums over the interval \(x\leq n\leq x+y\), with \(y\) being small relative to \(x\), thereby giving solutions to the ternary Goldbach problem with almost equal primes.
The authors now establish estimates for the sum \[ S=\sum_{x<n\leq x+y}\Lambda(n)e^{2\pi in^2\alpha}, \] valid for \(x^\theta\leq y\leq x\) with \(\theta>{11\over 16}\). Because of the nonlinear nature of \(n^2\), the sum \(S\) has a more complicated dominating term than that for the linear case when \(\alpha\) lies in the major arcs of the circle method. It is too complicated to state here these estimates, which depend on diophantine properties associated with \(\alpha\), but they can be applied to show that, for any fixed \(A>0\), \[ \sum_{x<n\leq x+y}\mu(n)e^{2\pi in^2\alpha}\ll y(\log x)^{-A}, \] uniformly in \(\alpha\) and in the same range of \(y\). The short interval can be reduced to \(\theta>{2\over 3}\) under the Generalised Riemann Hypothesis.
The main problem in the estimation of \(S\) lies in the case of the minor arcs, which have to be further subdivided into two sets in order to apply a combination of elementary and analytic methods, and the argument involves the use of Weyl’s method and Vaughan’s identity.

MSC:

11L07 Estimates on exponential sums
11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
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