**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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