The Number of Representations of an Integer by a Quadratic Form

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Khajehpour, Davood | 2018

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51050 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Pournaki, Mohammad Reza; Rajaei, Ali
Abstract:
In this paper Alexander Berkovich and Hamza Yesilyurt revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x2 + 5y2. Making use of Ramanujan’s 1 1 summation formula, they establish a new Lambert series identity for Σ1 n;m=1 qn2+5m2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But they do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, they 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, they find many new multiplicative eta-quotients and determine their coefficients
Keywords:
Quadratic Forms ; Gaussian Composition ; Lambert Series