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 x 2+5y 2. Making use of Ramanujan’s 1 ψ 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 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 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.