Lost Notebook · Dirichlet series
On two open points in Part IV of Ramanujan's Lost Notebook: the domain of Entry 12.2.3 and a summation reading of Entry 19.2.2
\[\sigma_c\!\left(\sum_{n\ge1} \chi(n)\,d(n)\,n^{-s}\right) \in \left[\tfrac14,\ \tfrac13\right]\]
The expected domain of Entry 12.2.3 fails for r = 0: the series diverges for 0 < Re s < 1/4.
- \(\chi = \text{non-principal character mod } 4\)
- \(L(s,\chi)\,L(s-r,\chi) = \sum \chi(n)\,\sigma_r(n)\,n^{-s}\)
Abstract
A short note on two places in Andrews and Berndt, Ramanujan's Lost Notebook, Part IV, where a question is left open. For Entry 12.2.3 the expected domain fails for r = 0: the abscissa of convergence of Σ χ(n)d(n)n^(−s) lies in [1/4, 1/3]. For Entry 19.2.2 a summation reading is given.
Cite
Otakhon U. Kenjaev (2026). On two open points in Part IV of Ramanujan's Lost Notebook: the domain of Entry 12.2.3 and a summation reading of Entry 19.2.2. Zenodo. https://doi.org/10.5281/zenodo.23003573
@misc{kenjaev2026lostnotebookiventries,
author = {Kenjaev, Otakhon U.},
title = {On two open points in Part IV of Ramanujan's Lost Notebook: the domain of Entry 12.2.3 and a summation reading of Entry 19.2.2},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.23003573}
}