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
Received: 30 November 2015
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.
Author address:
[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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