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Exponential sums and additive problems involving square-free numbers. (English) Zbl 1019.11028

Let \(r_{\nu}(N)\) denote the number of representations of \(N\) as the sum of \(\nu\) square-free numbers. Asymptotic formulae for \(r_{\nu}(N)\) with \(\nu\geq 2\) were obtained by Evelyn and Linfoot in their series of papers around 1930, and the paper under review is devoted to research on error terms in the formulae which are defined by \[ E_{\nu}(N)=r_{\nu}(N)-\frac{N^{\nu-1}}{(\nu-1)!} \Bigl(\frac 6{\pi^2}\Bigr)^{\nu} \prod_{p^2\nmid n}(1-(1-p^2)^{-\nu}) \prod_{p^2|n}(1-(1-p^2)^{1-\nu}). \] The authors prove the following results (in which \(\varepsilon\) denotes any fixed positive number as usual);
(i) \(E_{\nu}(N)\ll N^{\nu-3/2+\varepsilon}\) for \(\nu\geq 3\),
(ii) on writing \(\vartheta\) for the supremum of the real parts of the zeros of the Riemann zeta-function, one has \(E_{\nu}(N)=\Omega(N^{\nu-2+\vartheta/2-\varepsilon})\) for \(\nu\geq 2\),
(iii) under assumption of the generalized Riemann hypothesis for all Dirichlet \(L\)-functions, one has \(E_{\nu}(N)\ll N^{\nu-7/4+\varepsilon}\) for \(\nu\geq 4\), and \(E_3(N)\ll N^{37/28+\varepsilon}\).
The first theorem (i) improves the previously best bounds due to Mirsky, and is practically best possible in view of (ii) and the current state of knowledge of the distribution of zeros of the Riemann zeta-function. By (ii), moreover, the conditional result (iii) is best possible for \(\nu\geq 4\). These results may be extended to sums of \(k\)-free numbers with little extra effort.
The proof of (i) uses the circle method and is based on the following new result amongst others; for \(1\leq Q\leq N^{3/7}\), one has \[ \sup_{\alpha\in m(Q)}\Biggl|\sum_{n\leq N}\mu(n)^2e^{2\pi i\alpha n}\Biggr|\ll N^{1+\varepsilon}Q^{-1}, \] where \(\mu\) denotes the Möbius function, and \(m(Q)\) denotes the set of real numbers \(\alpha\in[0,1]\) such that there exist no integers \(q\) and \(a\) satisfying \(1\leq q\leq Q\) and \(|q\alpha-a|\leq Q/N\).

MSC:

11P55 Applications of the Hardy-Littlewood method
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11L07 Estimates on exponential sums
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References:

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