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On Averages of Exponential Sums over Primes

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Analytic Number Theory and Diophantine Problems

Part of the book series: Progress in Mathematics ((PM,volume 70))

Abstract

In this paper we shall be concerned with obtaining approximations to and estimates for the sum

$${{\text{S}}_{\text{N}}}(\alpha ){\text{ = }}\sum\limits_{{\text{n}} \leqslant {\text{N}}} {{\text{e}}({\text{n}}\alpha ) \wedge {\text{(n)}}}$$

where e(x) = exp(2πix), α is real, and Λ(n) is the von Mangoldt function. Although we are unable to establish the naturally conjectured results for this sum, we shall show how the introduction of averaging — in a form likely to occur in applications — can lead to substantial improvements.

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Author information

Authors and Affiliations

  1. Department of Pure Mathemtics, University College, P.O. Box 78, CF1 1XL, Cardiff, Wales, UK

    Glyn Harman

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  1. Glyn Harman
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Oklahoma State University, 74078, Stillwater, OK, USA

    A. C. Adolphson , J. B. Conrey  & A. Ghosh ,  & 

  2. Macquarie University, 2113, New South Wales, Australia

    R. I. Yager

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© 1987 Birkhäuser Boston

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Harman, G. (1987). On Averages of Exponential Sums over Primes. In: Adolphson, A.C., Conrey, J.B., Ghosh, A., Yager, R.I. (eds) Analytic Number Theory and Diophantine Problems. Progress in Mathematics, vol 70. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4816-3_13

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  • DOI: https://doi.org/10.1007/978-1-4612-4816-3_13

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-9173-2

  • Online ISBN: 978-1-4612-4816-3

  • eBook Packages: Springer Book Archive

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