Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms
Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms
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DOI：
10.1007/s11139-009-9215-8
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发表时间：
2006-11
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影响因子：
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通讯作者：
A. Berkovich;Hamza Yesilyurt
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文献类型：
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作者：
A. Berkovich;Hamza Yesilyurt
我们重温了费马和欧拉关于用二进制二次型x2 + 5 y2表示整数的旧命题。利用Ramanujan的1 <$1求和公式，建立了一个新的Lambert级数恒等式.费马和欧拉的猜想很容易遵循这个新公式。但我们不会就此止步。采用各种公式中发现拉马努金的笔记本电脑和使用有点巧妙，我们获得了一个新的朗伯系列收集某些无限产品与二次形式，如asx 2 + 6 y2，2x2+ 3 y2，x2+ 15 y2，3x 2 + 5 y2，x2+ 27 y2，x2+5（y2+z2+w2），5x 2 +y2+z2+w2。在这个过程中，我们发现了许多新的乘子，并确定了它们的系数。
We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic formx2+5y2. Making use of Ramanujan’s1ψ1summation formula, we establish a new Lambert series identity for. 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 asx2+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 multiplicativeeta-quotients and determine their coefficients.