{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:25:26Z","timestamp":1760149526039,"version":"build-2065373602"},"reference-count":104,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T00:00:00Z","timestamp":1692835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects\u2014Euler sums and series involving the zeta functions\u2014display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis\u2019s infinite product formula for \u03c0, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums.<\/jats:p>","DOI":"10.3390\/sym15091637","type":"journal-article","created":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T08:25:33Z","timestamp":1692951933000},"page":"1637","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Applications of Euler Sums and Series Involving the Zeta Functions"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7240-7737","authenticated-orcid":false,"given":"Junesang","family":"Choi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1277-8296","authenticated-orcid":false,"given":"Anthony","family":"Sofo","sequence":"additional","affiliation":[{"name":"College of Engineering and Science, Victoria University, P.O. Box 14428, Footscray, VIC 3011, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Berndt, B.C. (1985). Ramanujan\u2019s Notebooks, Springer. Part I.","DOI":"10.1007\/978-1-4612-1088-7"},{"key":"ref_2","first-page":"28","article-title":"On some infinite series","volume":"21","author":"Briggs","year":"1955","journal-title":"Scr. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier.","DOI":"10.1016\/B978-0-12-385218-2.00002-5"},{"key":"ref_4","first-page":"1191","article-title":"On an intriguing integral and some series related to \u03b6(4)","volume":"123","author":"Borwein","year":"1995","journal-title":"Proc. Am. Math. 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