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arXiv · 2010.10689

On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function

Matthew S. Fox·Chaitanya Karamchedu
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Abstract

We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on $\mathbb{R}$.

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Matthew S. Fox, Chaitanya Karamchedu. 2020-10-21. On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function. https://doi.org/10.1007/s11139-021-00412-z

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