{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:37:09Z","timestamp":1760236629080,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T00:00:00Z","timestamp":1639353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004704","name":"National Research Council of Thailand","doi-asserted-by":"publisher","award":["563NRCT322800"],"award-info":[{"award-number":["563NRCT322800"]}],"id":[{"id":"10.13039\/501100004704","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010034","name":"Walailak University","doi-asserted-by":"publisher","award":["563NRCT322800"],"award-info":[{"award-number":["563NRCT322800"]}],"id":[{"id":"10.13039\/501100010034","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunk\u0161tis, Kalpokas, and, more recently, Sowa. Whereas Sowa\u2019s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.<\/jats:p>","DOI":"10.3390\/sym13122410","type":"journal-article","created":{"date-parts":[[2021,12,14]],"date-time":"2021-12-14T01:22:05Z","timestamp":1639444925000},"page":"2410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann\u2019s Zeta-Function"],"prefix":"10.3390","volume":"13","author":[{"given":"Janyarak","family":"Tongsomporn","sequence":"first","affiliation":[{"name":"School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saeree","family":"Wananiyakul","sequence":"additional","affiliation":[{"name":"School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00f6rn","family":"Steuding","sequence":"additional","affiliation":[{"name":"Department of Mathematics, W\u00fcrzburg University, Am Hubland, 97218 W\u00fcrzburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"23","DOI":"10.4064\/aa-28-1-23-68","article-title":"Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory","volume":"28","author":"Berndt","year":"1975","journal-title":"Acta Arith."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1238","DOI":"10.1093\/imamat\/hxx033","article-title":"Riemann\u2019s zeta function and the broadband structure of pure harmonics","volume":"82","author":"Sowa","year":"2017","journal-title":"IMA J. Appl. Math."},{"key":"ref_3","first-page":"139","article-title":"Zeta zeros, Hurwitz zeta functions and L(1,\u03c7)","volume":"65","author":"Fujii","year":"1989","journal-title":"Proc. Jpn. Acad. Ser. Math. Sci."},{"key":"ref_4","first-page":"209","article-title":"Sum of the periodic zeta-function over the nontrivial zeros of the Riemann zeta-function","volume":"28","author":"Kalpokas","year":"2008","journal-title":"Analysis"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Laurin\u010dikas, A., and Garunk\u0161tis, R. (2002). The Lerch Zeta-Function, Springer Science & Business Media.","DOI":"10.1007\/978-94-017-6401-8"},{"key":"ref_6","unstructured":"Titchmarsh, E.C., and Heath-Brown, D.R. (1986). The Theory of the Riemann Zeta-Function, Oxford University Press. [2nd ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF01403094","article-title":"Mean values of the Riemann zeta-function and its derivatives","volume":"75","author":"Gonek","year":"1984","journal-title":"Invent. Math."},{"key":"ref_8","first-page":"129","article-title":"Representation of an odd number as the sum of three primes","volume":"15","author":"Vinogradov","year":"1937","journal-title":"Dokl. Akad. Nauk SSSR"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2410\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:46:35Z","timestamp":1760168795000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,13]]},"references-count":8,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122410"],"URL":"https:\/\/doi.org\/10.3390\/sym13122410","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,13]]}}}