{"paper":{"title":"On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Aman Singh, Rahul Kumar","submitted_at":"2026-05-20T13:31:37Z","abstract_excerpt":"In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series. More precisely, they conjectured that \\[ \\coeff{q^{N2^a}} \\sum_{m,k\\geq 1} \\frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} =\\sigma_1(N), \\] where $\\sigma_1(N)$ is the sum of all the positive divisors of $N$. The main idea of the proof is to first transform a double Lambert series on the left-hand side into a single sum. This leads us to derive a new representation of quasi-modular forms $E_2(q)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21163","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.21163/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}