Lost Notebook · Algebraic number theory
Pisot units and the third manuscript on page 343 of Ramanujan's Lost Notebook, with a verification script and calculator
\[\frac{1}{2^{1/5}-1} = 1+\theta+\theta^2+\theta^3+\theta^4,\qquad \theta = 2^{1/5}\]
Claims that Andrews and Berndt called wrong become a theorem about units of \(\mathbb{Z}[2^{1/5}]\).
- \(b = \frac{\sqrt5}{(1+4^{1/5})^{5/2}} = 1-\theta-\theta^2+\theta^4\)
- \(\tfrac{1}{2^{1/p}-1}\ \text{is Pisot} \iff p \le 6\)
Abstract
Andrews and Berndt (Ramanujan's Lost Notebook, Part IV, Sect. 7.4) could not give a meaning to Ramanujan's claims on page 343. We show that a = 1/(2^(1/5) − 1) and b = √5/(1 + 4^(1/5))^(5/2) are units of Z[2^(1/5)], that each factor of Ramanujan's error term is the modulus of a Galois conjugate of a^m b^n, and that the integer is a trace, which proves (7.4.1) with a sharp constant.
Cite
Otakhon U. Kenjaev (2026). Pisot units and the third manuscript on page 343 of Ramanujan's Lost Notebook, with a verification script and calculator. Zenodo. https://doi.org/10.5281/zenodo.23003128
@misc{kenjaev2026ramanujanpage343,
author = {Kenjaev, Otakhon U.},
title = {Pisot units and the third manuscript on page 343 of Ramanujan's Lost Notebook, with a verification script and calculator},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.23003128}
}