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

Details 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

2006-11-10

arxiv.org/abs/math/0611300v2

Mathematics

Number Theory

26 pages, no figures, fun to read

Scientific paper

We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_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 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 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.

Affiliated with

Also associated with

No associations

Rating

Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-566311