{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T05:23:42Z","timestamp":1781241822557,"version":"3.54.1"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"246","license":[{"start":{"date-parts":[[2004,11,26]],"date-time":"2004-11-26T00:00:00Z","timestamp":1101427200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The flow of the Riemann zeta function,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ModifyingAbove s With dot equals zeta left-parenthesis s right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>s<\/mml:mi>\n                                <mml:mo>\n                                  \u02d9\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\dot {s}=\\zeta (s)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica. The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01529-1","type":"journal-article","created":{"date-parts":[[2004,4,21]],"date-time":"2004-04-21T12:29:08Z","timestamp":1082550548000},"page":"987-1004","source":"Crossref","is-referenced-by-count":11,"title":["The holomorphic flow of the Riemann zeta function"],"prefix":"10.1090","volume":"73","author":[{"given":"Kevin","family":"Broughan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Barnett","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2003,11,26]]},"reference":[{"key":"1","series-title":"Undergraduate Texts in Mathematics","volume-title":"Introduction to analytic number theory","author":"Apostol, Tom M.","year":"1976"},{"issue":"1899","key":"2","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1098\/rspa.1992.0053","article-title":"A new asymptotic representation for \ud835\udf01(\\frac12+\ud835\udc56\ud835\udc61) and quantum spectral determinants","volume":"437","author":"Berry, M. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"5","isbn-type":"print","first-page":"29","article-title":"An efficient algorithm for the Riemann zeta function","author":"Borwein, P.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0821821679"},{"key":"6","doi-asserted-by":"crossref","unstructured":"K. A. Broughan, Holomorphic flows on simply connected domains have no limit cycle, Meccanica 38(6) (2003), 699\u2013709.","DOI":"10.1023\/A:1025821123532"},{"key":"7","unstructured":"K. A. Broughan, http:\/\/www.math.waikato.ac.nz\/~kab, Web site for zeta phase portraits and data for the zeros encoding."},{"key":"8","doi-asserted-by":"publisher","first-page":"611","DOI":"10.2307\/1968946","article-title":"On Frobeniusean algebras. I","volume":"40","author":"Nakayama, Tadasi","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"9","series-title":"Royal Society Mathematical Tables, Vol. 6","volume-title":"Tables of the Riemann zeta function","author":"Haselgrove, C. B.","year":"1960"},{"key":"10","isbn-type":"print","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1090\/conm\/290\/04575","article-title":"Computing the Riemann zeta function by numerical quadrature","author":"Galway, William F.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0821820796"},{"key":"11","unstructured":"P. Godfrey, An efficient algorithm for the Riemann zeta function, http:\/\/www.mathworks. com\/support\/ftp zeta.m, etan.m (2000)"},{"key":"12","doi-asserted-by":"crossref","unstructured":"U. D. Jentschura, P. J. Mohr, G. Soff, E. J. Weniger, Convergence acceleration via combined nonlinesr-condensation transformations, Computer Physics Comm. 116 (1999), 28-54.","DOI":"10.1016\/S0010-4655(98)00111-8"},{"issue":"174","key":"13","doi-asserted-by":"publisher","first-page":"667","DOI":"10.2307\/2008005","article-title":"On the zeros of the Riemann zeta function in the critical strip. IV","volume":"46","author":"van de Lune, J.","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"797","DOI":"10.2307\/2000939","article-title":"Fast algorithms for multiple evaluations of the Riemann zeta function","volume":"309","author":"Odlyzko, A. M.","year":"1988","journal-title":"Trans. Amer. Math. 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