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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL, 32611, USA

    Alexander Berkovich

  2. Department of Mathematics, Bilkent University, 06800, Bilkent/Ankara, Turkey

    Hamza Yesilyurt

Authors
  1. Alexander Berkovich
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  2. Hamza Yesilyurt
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Correspondence to Alexander Berkovich.

Additional information

Research of the first author was supported in part by the NSA Grant H98230-07-1-0011.

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Berkovich, A., Yesilyurt, H. Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms. Ramanujan J 20, 375–408 (2009). https://doi.org/10.1007/s11139-009-9215-8

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  • DOI: https://doi.org/10.1007/s11139-009-9215-8

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