Timeline for Two versions of the Möbius inversion formula
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 18, 2018 at 19:43 | vote | accept | 352506 | ||
| Dec 17, 2018 at 20:23 | answer | added | Will Sawin | timeline score: 1 | |
| Dec 10, 2018 at 18:49 | comment | added | KConrad | I don't know what proof you have in mind for the second version, but all you're doing to define $g$ in terms of $f$ is adding values of $f$, hence only the additive structure is relevant. Does your second proof really involve multiplying elements of $R$ together where neither term is an integer? | |
| Dec 10, 2018 at 14:38 | comment | added | Andreas Blass | The convolution doesn't seem to use the multiplicative part of the ring structure. You're convolving a sequence $g$ of elements of $R$ with a sequence $\mu$ of natural numbers, and that makes sense in any additive abelian group. | |
| Dec 10, 2018 at 14:16 | comment | added | 352506 | Perhaps I should reformulate my question in terms of the proofs of the formulas. The proof of the second version uses the multiplicative structure of $R$ (via the convolution). Is there an analogous proof of the first version? | |
| Dec 10, 2018 at 2:35 | comment | added | KConrad | The second version never uses the ring structure in an essential way. It is formulated only with the additive group structure of ${\mathcal A}(R)$, so version 2 is really a special case of version 1. The "common generalization" is just version 1. | |
| Dec 10, 2018 at 2:01 | history | asked | 352506 | CC BY-SA 4.0 |