{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:36:23Z","timestamp":1776800183480,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"298","license":[{"start":{"date-parts":[[2016,10,15]],"date-time":"2016-10-15T00:00:00Z","timestamp":1476489600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-0932078"],"award-info":[{"award-number":["DMS-0932078"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1406190"],"award-info":[{"award-number":["DMS-1406190"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","award":["DMS-0932078"],"award-info":[{"award-number":["DMS-0932078"]}],"id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","award":["DMS-1406190"],"award-info":[{"award-number":["DMS-1406190"]}],"id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -functions to a powerfull modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable the square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control. The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor. Practical parameter choices are discussed.\n                  <\/p>","DOI":"10.1090\/mcom\/3019","type":"journal-article","created":{"date-parts":[[2015,10,15]],"date-time":"2015-10-15T09:09:06Z","timestamp":1444900146000},"page":"1017-1032","source":"Crossref","is-referenced-by-count":2,"title":["An alternative to Riemann-Siegel type formulas"],"prefix":"10.1090","volume":"85","author":[{"given":"Ghaith","family":"Hiary","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2015,10,15]]},"reference":[{"issue":"274","key":"1","doi-asserted-by":"publisher","first-page":"995","DOI":"10.1090\/S0025-5718-2010-02426-3","article-title":"High precision computation of Riemann\u2019s zeta function by the Riemann-Siegel formula, I","volume":"80","author":"Arias de Reyna, J.","year":"2011","journal-title":"Math. 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Gabcke, Neue herleitung und explicite restabsch\u00e4tzung der riemann-siegel-formel., Ph.D. thesis, G\u00f6ttingen, 1979."},{"key":"8","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1016\/j.jnt.2013.12.005","article-title":"Computing Dirichlet character sums to a power-full modulus","volume":"140","author":"Hiary, Ghaith A.","year":"2014","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"891","DOI":"10.4007\/annals.2011.174.2.4","article-title":"Fast methods to compute the Riemann zeta function","volume":"174","author":"Hiary, Ghaith Ayesh","year":"2011","journal-title":"Ann. of Math. 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