Ramanujan’s Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms
نویسندگان
چکیده
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x2 + 5y2. Making use of Ramanujan’s 1ψ1 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 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, we find many new multiplicative eta-quotients and determine their coefficients.
منابع مشابه
2-universal Hermitian Lattices over Imaginary Quadratic Fields
We call a positive definite integral quadratic form universal if it represents all positive integers. Then Lagrange’s Four Square Theorem means that the sum of four squares is universal. In 1930, Mordell [M] generalized this notion to a 2-universal quadratic form: a positive definite integral quadratic form that represents all binary positive definite integral quadratic forms, and showed that t...
متن کاملSimple Proofs for Universal Binary Hermitian Lattices
It has been a central problem in the theory of quadratic forms to find integers represented by quadratic forms. The celebrated Four Square Theorem by Lagrange [10] was an outstanding result in this study. Ramanujan generalized this theorem and found 54 positive definite quaternary quadratic forms which represent all positive integers [13]. We call a positive definite quadratic form universal, i...
متن کامل2-universal Positive Definite Integral Quinary Diagonal Quadratic Forms
As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms. §
متن کاملEuler Products in Ramanujan’s Lost Notebook
In his famous paper, “On certain arithmetical functions”, Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan’s tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for...
متن کامل2-universal Positive Definite Integral Quinary Quadratic Forms
As a generalization of the famous four square theorem of Lagrange, Ramanujan and Willerding found all positive definite integral quaternary quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary quadratic forms that represent all positive definite integral binary quadratic forms. We also discuss recent results on positive definite int...
متن کامل