Ramanujan ' s identities and representation of integers by certain binary and quaternary quadratic forms

Abstract

We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x2+5y2. Making use of Ramanujan's 1ψ1 summation formula, we establish a new Lambert series identity for Σ∞n,m=-∞qn2+5m2 Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we do not stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity, we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x2+6y2, 2x2+3y2, x2+15y2, 3x2+5y2, x2+27y2, x2+5(y2+z2+w2), 5x2+y2+z2+w2. In the process, we find many new multiplicative eta-quotients and determine their coefficients. © 2009 Springer Science+Business Media, LLC.