We obtain tight bounds on the L^1 norm of the exponential sum $M(\alpha) = \sum_{n\le X}\tau(n)e(n\alpha)$ where $\tau(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\gg \sqrt{X}\log X$.
The $L^1$ mean of the exponential sum of the divisor function
On the $L^1$ norm of an exponential sum involving the divisor function
Moments of an exponential sum related to the divisor function
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