{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T01:51:29Z","timestamp":1775872289135,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T00:00:00Z","timestamp":1761004800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper we will be concerned with zeta-symmetry\u2014the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)\u2014and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan\u2019s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle).<\/jats:p>","DOI":"10.3390\/axioms14100774","type":"journal-article","created":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T15:36:31Z","timestamp":1761060991000},"page":"774","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Cotangent Function as an Avatar of the Polylogarithm Function of Order 0 and Ramanujan\u2019s Formula"],"prefix":"10.3390","volume":"14","author":[{"given":"Ruiyang","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics, NorthWest University, Xi\u2019an 710069, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haoyang","family":"Lu","sequence":"additional","affiliation":[{"name":"Taishan College, Shandong University, Shanda Nanlu 27, Jinan 250110, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2131-1962","authenticated-orcid":false,"given":"Shigeru","family":"Kanemitsu","sequence":"additional","affiliation":[{"name":"SUDA Reseach Institute, No. 1, Taiyang Road, Sanmenxia Economic Development Zone, Sanmenxia 472000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1216\/RMJ-1977-7-1-147","article-title":"Modular transformations and generalization of several formulas of Ramanujan","volume":"7","author":"Berndt","year":"1977","journal-title":"Rocky Mount. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Berndt, B.C. (1985). Ramanujan\u2019s Notebooks Part I, Springer.","DOI":"10.1007\/978-1-4612-1088-7"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chakraborty, K., Kanemitsu, S., and Kuzumaki, T. (2025). Modular Relations and Parity in Number Theory, Springer Nature.","DOI":"10.1007\/978-981-96-6471-9"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kanemitsu, S., Tanigawa, Y., and Yoshimoto, M. (2002). Ramanujan\u2019s formula and modular forms. Number-Theoretic Methods-Future Trends, Proceedings of the Second China-Japan Seminar, Tokyo, Japan, 21\u201323 September 2002, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4757-3675-5_10"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Serre, J.-P. (1973). 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Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/774\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T15:58:10Z","timestamp":1761062290000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/774"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,21]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["axioms14100774"],"URL":"https:\/\/doi.org\/10.3390\/axioms14100774","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,21]]}}}