Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

Alessandro Languasco[1]

Applications of some exponential sums on prime powers: a survey

Pages: 19-37
Accepted in revised form: 17 May 17 2016
Mathematics Subject Classification (2010): Primary 11P32; Secondary 11P55, 11P05, 44A10, 33C10.
Keywords: Waring-Goldbach problem, Hardy-Littlewood method, Laplace transforms, Cesàro averages.
[1] : Università di Padova, Dipartimento di Matematica, Via Trieste 63, Padova, 35121, Italy

This research was partially supported by the grant PRIN2010-11 Arithmetic Algebraic Geometry and Number Theory.

Abstract: Let $$\Lambda$$ be the von Mangoldt function and $$N,\ell\geq 1$$ be two integers. We will see some results by the author and Alessandro Zaccagnini obtained using the original Hardy & Littlewood circle method function, i.e.

$$\widetilde{S}_{\ell}(\alpha) = \sum_{n=1}^{\infty} \Lambda(n) e^{-n^{\ell}/N} e(n^{\ell}\alpha),$$

where $$e(x)=\exp(2\pi i x)$$, instead of $$S_{\ell}(\alpha) = \sum_{n=1}^{N} \Lambda(n) e(n^{\ell}\alpha)$$. We will also motivate why, for some short interval additive problems, the approach using $$\widetilde{S}_{\ell}(\alpha)$$ gives sharper results than the ones that can be obtained with $$S_{\ell}(\alpha)$$. The final section of this paper is devoted to correct an oversight occurred in [17] and [19].

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