On the mean value of the magnitude of an exponential sum involving the divisor function

Abstract

We obtain an asymptotic formula for 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 $\sim C\sqrt{X}\log X$ with $C=\frac{18}{{\pi }^3}-\frac{12\log 2}{{\pi }^3}-\frac{1}{2\pi }\approx 0.153\dots$.

There is an error in the final section in the computation of the main term. When this is corrected, one obtains that the main term is actually $\ll \sqrt{X}$. The result one obtains with the methods of this paper is then that the L^1 norm is $O(\sqrt{X})$