{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:29Z","timestamp":1776721949895,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We describe computations which show that each of the first 12069 zeros of the Ramanujan\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c4\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -Dirichlet series of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma plus i t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma + i t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the region\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 greater-than t greater-than 6397\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>6397<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 &gt; t &gt; 6397<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is simple and lies on the line\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma equals 6\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>6<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma = 6<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The failures of Gram\u2019s law in this region are also noted. The first\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5018\">\n                        <mml:semantics>\n                          <mml:mn>5018<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">5018<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    zeros and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2228\">\n                        <mml:semantics>\n                          <mml:mn>2228<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">2228<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    successive zeros beginning with the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"20001\">\n                        <mml:semantics>\n                          <mml:mn>20001<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">20001<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    st zero are also calculated. The distribution of the normalized spacing of the zeros is examined and it appears to be that of the eigenvalues of random matrices. These comptuations are done with a Berry-Keating formula for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c4\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -Dirichlet series and evaluated using\n                    <italic>Mathematica<\/italic>\n                    \u2122.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00734-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1613-1619","source":"Crossref","is-referenced-by-count":2,"title":["On the zeros of the Ramanujan \ud835\udf0f-Dirichlet series in the critical strip"],"prefix":"10.1090","volume":"65","author":[{"given":"J.","family":"Keiper","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"1899","key":"1","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. V.","year":"1992","journal-title":"Proc. Roy. Soc. London Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0962-8444","issn-type":"print"},{"key":"2","series-title":"Pure and Applied Mathematics, Vol. 58","volume-title":"Riemann's zeta function","author":"Edwards, H. M.","year":"1974"},{"issue":"165","key":"3","doi-asserted-by":"publisher","first-page":"241","DOI":"10.2307\/2007577","article-title":"On zeros of Mellin transforms of \ud835\udc46\ud835\udc3f\u2082(\ud835\udc4d) cusp forms","volume":"42","author":"Ferguson, H. R. P.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","unstructured":"A. Guthmann, Ramanujans Tau-Funktion, unpublished, June, 1988."},{"key":"5","volume-title":"Ramanujan: twelve lectures on subjects suggested by his life and work","author":"Hardy, G. H.","year":"1959"},{"issue":"198","key":"6","doi-asserted-by":"publisher","first-page":"765","DOI":"10.2307\/2153215","article-title":"Power series expansions of Riemann\u2019s \ud835\udf09 function","volume":"58","author":"Keiper, J. B.","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"177","key":"7","doi-asserted-by":"publisher","first-page":"273","DOI":"10.2307\/2007890","article-title":"On the distribution of spacings between zeros of the zeta function","volume":"48","author":"Odlyzko, A. M.","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","unstructured":"A. M. Odlyzko, The 10\u00b2\u2070-th Zero of the Riemann Zeta function and 70 Million of its Neighbors, unpublished."},{"key":"9","doi-asserted-by":"publisher","first-page":"379","DOI":"10.2307\/2005626","article-title":"Calculation of the Ramanujan \ud835\udf0f-Dirichlet series","volume":"27","author":"Spira, Robert","year":"1973","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"10","first-page":"87","article-title":"On calculations of zeros of \ud835\udc3f-functions related with Ramanujan\u2019s discriminant function on the critical line","volume":"3","author":"Yoshida, Hiroyuki","year":"1988","journal-title":"J. Ramanujan Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0970-1249","issn-type":"print"},{"key":"11","unstructured":"H. Yoshida, On Calculations of Zeros of Various L-functions, to appear."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00734-X\/S0025-5718-96-00734-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00734-X\/S0025-5718-96-00734-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:17:19Z","timestamp":1776719839000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00734-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":11,"journal-issue":{"issue":"216","published-print":{"date-parts":[[1996,10]]}},"alternative-id":["S0025-5718-96-00734-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00734-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996]]}}}