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On exponential sums involving the Ramanujan function

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Abstract

Let τ(n) be the arithmetical function of Ramanujan, α any real number, and x≥2. The uniform estimate

$$\mathop \Sigma \limits_{n \leqslant x} \tau (n)e(n\alpha ) \ll x^6 \log x$$

is a classical result of J R Wilton. It is well known that the best possible bound would be ≪x 6. The validity of this hypothesis is proved.

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, SF-20500, Turku, Finland

    M. Jutila

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  1. M. Jutila
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Jutila, M. On exponential sums involving the Ramanujan function. Proc. Indian Acad. Sci. (Math. Sci.) 97, 157–166 (1987). https://doi.org/10.1007/BF02837820

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  • DOI: https://doi.org/10.1007/BF02837820

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