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

TL;DR

This paper derives an asymptotic formula for the L^1 norm of an exponential sum involving the divisor function, revealing its growth rate as approximately 0.153 times .5 times log X.

Contribution

It provides the first precise asymptotic estimate for the mean value of an exponential sum with the divisor function.

Findings

L^1 norm asymptotically .153 .5 imes ext{log} X

Confirmed the growth rate of the sum's magnitude

Established a constant factor for the asymptotic formula

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 $C = \frac{18}{\pi^3} - \frac{12\log 2}{\pi^3} - \frac{1}{2\pi}\approx 0.153\dots$.