Lost Notebook · Lattice sums
The constants K and λ in Ramanujan's identity (9.2.5) of the Lost Notebook, with a hexagonal lattice sum calculator
\[K = \mu-\nu,\qquad \lambda = \mu+\nu \qquad (\gcd(\mu,\nu)=1)\]
Andrews and Berndt could not identify K and λ. Six-fold symmetry of the Eisenstein integers settles them.
- \(R(6k) = \tfrac{(-1)^k}{3}\left(c_k\,E_6(\rho)^k - 3\right)\)
- \(c_1 = 1,\ c_2 = \tfrac{250}{691},\ c_3 = \tfrac{5500}{43867}\)
Abstract
Andrews and Berndt (Ramanujan's Lost Notebook, Part IV, pp. 215-216) could not identify the constants K and λ in Ramanujan's identity (9.2.5). We show that it holds for Re s > 2 with K = μ − ν and λ = μ + ν (sum over coprime μ, ν ≥ 1), prove it via the six-fold symmetry of the Eisenstein integers, and give an in-browser calculator for the resulting hexagonal sector lattice sum.
Cite
Otakhon U. Kenjaev (2026). The constants K and λ in Ramanujan's identity (9.2.5) of the Lost Notebook, with a hexagonal lattice sum calculator. Zenodo. https://doi.org/10.5281/zenodo.22987480
@misc{kenjaev2026ramanujanidentity925,
author = {Kenjaev, Otakhon U.},
title = {The constants K and λ in Ramanujan's identity (9.2.5) of the Lost Notebook, with a hexagonal lattice sum calculator},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.22987480}
}