{"paper":{"title":"On a secant Dirichlet series and Eichler integrals of Eisenstein series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Armin Straub, Bruce C. Berndt","submitted_at":"2014-06-09T18:21:49Z","abstract_excerpt":"We consider, for even $s$, the secant Dirichlet series $\\psi_s (\\tau) = \\sum_{n = 1}^{\\infty} \\frac{\\sec (\\pi n \\tau)}{n^s}$, recently introduced and studied by Lal\\'{\\i}n, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lal\\'{\\i}n, Rodrigue and Rogers, that the values $\\psi_{2 m} ( \\sqrt{r})$, with $r > 0$ rational, are rational multiples of $\\pi^{2 m}$. We then put the properties of the secant Dirichlet series into context by showing that they are Eichler integrals of odd weight Eisenstein series of level $4$. This leads us to consider Eichler integrals of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.2273","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":""},"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"}