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

Mayank Pandey

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arXiv · 1901.07280

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

Mayank Pandey

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Abstract

We obtain an asymptotic formula for the L^1 norm of the exponential sum $M(α) = \sum_{n\le X}τ(n)e(nα)$ where $τ(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\sim C\sqrt{X}\log X$ with $C = \frac{18}{π^3} - \frac{12\log 2}{π^3} - \frac{1}{2π}\approx 0.153\dots$.

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BibTeX RIS

Mayank Pandey. 2020-05-17. On the mean value of the magnitude of an exponential sum involving the divisor function. https://arxiv.org/abs/1901.07280

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