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arXiv · math/0611300

Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms

Alexander Berkovich·Hamza Yesilyurt
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Abstract

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.

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Alexander Berkovich, Hamza Yesilyurt. 2007-02-06. Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms. https://arxiv.org/abs/math/0611300

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