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

2006/11/10 by Alexander Bérkovich, Berkovich, Alexander, Hamza Yesilyurt +1
Mathematics · #05A19 #05A30 #11E16 #11E25 #11F27 #11F30 #11R29 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0611300

openalex publication_date 2006/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x²+5y². Making use of Ramanujan's₁ψ₁ summation formula we establish a new Lambert series identity for ∑_n,m=-∞^∞ q^n2+5m2. 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 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.

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