Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
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.
26 pages, no figures, fun to read
26 pages, no figures, fun to read