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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Notes
We use the asymptotic notation \(f = {\mathcal {O}}_x(g)\) to denote the estimate \(|f| \le C_x g\) for some absolute constant \(C_x > 0\) that may depend on the parameter x.
We explicitly include in \({\widetilde{\sigma }}_1(x, \alpha )\) a functional dependence on \(\alpha \) to highlight the fact that \(\alpha \) is implicit in the \(a_n\) coefficients.
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Fox, M.S., Karamchedu, C. On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function. Ramanujan J 58, 229–237 (2022). https://doi.org/10.1007/s11139-021-00412-z
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DOI: https://doi.org/10.1007/s11139-021-00412-z