{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:39:59Z","timestamp":1760240399812,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,4]],"date-time":"2019-06-04T00:00:00Z","timestamp":1559606400000},"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>In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindel\u00f6f hypothesis. Some of the above relations motivate the need for analysing the large    \u03b1    behaviour of the modified Hurwitz zeta function      \u03b6 1   ( s , \u03b1 )     ,     s \u2208 C    ,     \u03b1 \u2208 ( 0 , \u221e )    , which is also presented here.<\/jats:p>","DOI":"10.3390\/sym11060754","type":"journal-article","created":{"date-parts":[[2019,6,5]],"date-time":"2019-06-05T09:37:58Z","timestamp":1559727478000},"page":"754","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Relations among the Riemann Zeta and Hurwitz Zeta Functions, as Well as Their Products"],"prefix":"10.3390","volume":"11","author":[{"given":"A. C. L.","family":"Ashton","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK"}]},{"given":"A. S.","family":"Fokas","sequence":"additional","affiliation":[{"name":"Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089-2560, USA"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,4]]},"reference":[{"key":"ref_1","unstructured":"Fokas, A.S. (2018). A novel approach to the Lindel\u00f6f hypothesis. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1090\/S0273-0979-03-00995-9","article-title":"Riemann\u2019s zeta function and beyond","volume":"41","author":"Gelbart","year":"2003","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_3","first-page":"9","article-title":"A new method of estimation for trigonometrical sums","volume":"43","author":"Vinogradov","year":"1935","journal-title":"Mat. 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Novel identities for certain double exponential sums, in preparation."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.jmaa.2018.05.012","article-title":"Asymptotics to all orders of the Hurwitz zeta function","volume":"465","author":"Fernandez","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","unstructured":"Fokas, A.S., and Lenells, J. (2015). On the asymptotics to all orders of the Riemann zeta function and of a two parameter generalisation of the Riemann zeta function. arXiv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01457521","article-title":"On Hurwitz zeta-function","volume":"264","author":"Rane","year":"1983","journal-title":"Math. 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Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/754\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:56:03Z","timestamp":1760187363000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/754"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,4]]},"references-count":15,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060754"],"URL":"https:\/\/doi.org\/10.3390\/sym11060754","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,4]]}}}