On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function
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. W...
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| Veröffentlicht in: | The Ramanujan journal 2022-05, Vol.58 (1), p.229-237 |
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| container_title | The Ramanujan journal |
| container_volume | 58 |
| creator | Fox, Matthew S. Karamchedu, Chaitanya |
| description | 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
R
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| doi_str_mv | 10.1007/s11139-021-00412-z |
| format | Article |
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| subjects | Combinatorics Convergence Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Mathematical functions Mathematics Mathematics and Statistics Number Theory Real variables Sums |
| title | On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function |
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