Applications of some exponential sums on prime powers: a survey. (English) Zbl 1415.11127

Languasco, Alessandro

Riv. Mat. Univ. Parma (N.S.) 7, No. 1, 19-37 (2016).

Summary: 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 [the author and A. Zaccagnini, J. Math. Anal. Appl. 401, No. 2, 568–577 (2013; Zbl 1348.11071)] and [the author and A. Zaccagnini, Forum Math. 27, No. 4, 1945–1960 (2015; Zbl 1386.11105)].

MSC:

11P32	Goldbach-type theorems; other additive questions involving primes
11P55	Applications of the Hardy-Littlewood method
11P05	Waring’s problem and variants
44A10	Laplace transform
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Keywords:

Waring-Goldbach problem; Hardy-Littlewood method; Laplace transforms; Cesàro averages

Citations:

Zbl 1348.11071; Zbl 1386.11105