Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms
| dc.creator | Berkovich, Alexander | |
| dc.creator | Yesilyurt, Hamza | |
| dc.date | 2006-11-10 | |
| dc.date | 2007-02-06 | |
| dc.date.accessioned | 2026-07-07T07:44:46Z | |
| dc.date.available | 2026-07-07T07:44:46Z | |
| dc.description | We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1ψ_1 summation formula we establish a new Lambert series identity for \sum_{n,m=-\infty}^{\infty} q^{n^2+5m^2}. Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we don't stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity we obtaina collection of new Lambert series for certain infinite products associated with quadratic forms such as x^2+6y^2, 2x^2+3y^2, x^2+15y^2, 3x^2+5y^2, x^2+27y^2, x^2+5(y^2+ z^2+ w^2), 5x^2+y^2+ z^2+ w^2. In the process, we find many new multiplicative eta-quotients and determine their coefficients. | |
| dc.description | 26 pages, no figures, fun to read | |
| dc.identifier | https://arxiv.org/abs/math/0611300 | |
| dc.identifier | http://arxiv.org/abs/math/0611300 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123320 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11E16, 11E25, 11F27, 11F30; 05A19, 05A30, 11R29 | |
| dc.title | Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms | |
| dc.type | text |