Exponential sums over products and their \(L_1\)-norm (Q5937387)

For a natural number \(n\), let \(d(n)\) denote the number of (positive) divisors of \(n\). In this article, it is proved that \[ \sqrt{x}\ll \int_0^1\Bigl|\sum_{n\leq x}d(n)\exp(2\pi in\alpha)\Bigr|d\alpha \ll \sqrt{x} \] for real numbers \(x\geq 1\). The proof uses a Hardy-Littlewood dissection of the unit interval and is based on a natural expression of the above exponential sum as a double sum. As the author points out, the method of this paper is also applicable to show, for \(y\geq 1\), that \[ y\ll\int_0^1\Bigl|\sum_{u\leq y}\sum_{v\leq y}\exp(2\pi iuv\alpha)\Bigr|d\alpha\ll y . \]