{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:11:55Z","timestamp":1776863515680,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"302","license":[{"start":{"date-parts":[[2017,1,15]],"date-time":"2017-01-15T00:00:00Z","timestamp":1484438400000},"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                    We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q less-than-or-equal-to 400 000\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>400<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>000<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q\\leq 400\\,000<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We check, to height,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"max left-parenthesis StartFraction 10 Superscript 8 Baseline Over q EndFraction comma StartFraction upper A dot 10 Superscript 7 Baseline Over q EndFraction plus 200 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext>max<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mfrac>\n                                <mml:msup>\n                                  <mml:mn>10<\/mml:mn>\n                                  <mml:mn>8<\/mml:mn>\n                                <\/mml:msup>\n                                <mml:mi>q<\/mml:mi>\n                              <\/mml:mfrac>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mfrac>\n                                <mml:mrow>\n                                  <mml:mi>A<\/mml:mi>\n                                  <mml:mo>\n                                    \u22c5\n                                    \n                                  <\/mml:mo>\n                                  <mml:msup>\n                                    <mml:mn>10<\/mml:mn>\n                                    <mml:mn>7<\/mml:mn>\n                                  <\/mml:msup>\n                                <\/mml:mrow>\n                                <mml:mi>q<\/mml:mi>\n                              <\/mml:mfrac>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>200<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textrm {max}\\left (\\frac {10^8}{q},\\frac {A\\cdot 10^7}{q}+200\\right )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A equals 7.5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>7.5<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A=7.5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the case of even characters and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A equals 3.75\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3.75<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A=3.75<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for odd characters. In addition we confirm that no Dirichlet L-function with a modulus\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q less-than-or-equal-to 2 000 000\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>000<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>000<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q\\leq 2\\,000\\,000<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    vanishes at its central point.\n                  <\/p>","DOI":"10.1090\/mcom\/3077","type":"journal-article","created":{"date-parts":[[2015,5,20]],"date-time":"2015-05-20T10:40:43Z","timestamp":1432118443000},"page":"3009-3027","source":"Crossref","is-referenced-by-count":23,"title":["Numerical computations concerning the GRH"],"prefix":"10.1090","volume":"85","author":[{"given":"David","family":"Platt","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,1,15]]},"reference":[{"key":"1","series-title":"Undergraduate Texts in Mathematics","volume-title":"Introduction to analytic number theory","author":"Apostol, Tom M.","year":"1976"},{"key":"2","doi-asserted-by":"crossref","unstructured":"L. Bluestein, A linear filtering approach to the computation of discrete Fourier transform, IEEE Transactions on Audio and Electroacoustics, 18(4):451\u2013455, 1970.","DOI":"10.1109\/TAU.1970.1162132"},{"issue":"4","key":"3","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1080\/10586458.2006.10128976","article-title":"Artin\u2019s conjecture, Turing\u2019s method, and the Riemann hypothesis","volume":"15","author":"Booker, Andrew R.","year":"2006","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"4","isbn-type":"print","volume-title":"The DFT","author":"Briggs, William L.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713420"},{"key":"5","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/0022-247X(67)90183-7","article-title":"On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem","volume":"18","author":"Brown, J. L., Jr.","year":"1967","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"6","unstructured":"X. Gourdon, The 10\u00b9\u00b3 First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height, http:\/\/numbers.computation.free.fr\/Constants\/Miscellaneous\/ zetazeros1e13-1e24.pdf, 2010."},{"key":"7","unstructured":"H.A. Helfgott, Minor arcs for Goldbach\u2019s problem, arXiv preprint arXiv:1205.5252, 2012."},{"key":"8","unstructured":"H.A. Helfgott, Major arcs for Goldbach\u2019s problem,  arXiv preprint arXiv:1305.2897, 2013."},{"key":"9","unstructured":"IEEE, IEEE Standard for Binary Floating-Point Arithmetic, IEEE Std 754-1985., 1985."},{"key":"10","unstructured":"B. Lambov, Reliable Implementation of Real Number Algorithms: Theory and Practice, chapter Interval Arithmetic Using SSE-2, Lecture Notes in Computer Science. Springer, 2008."},{"key":"11","first-page":"61","article-title":"The automatic analysis and control of error in digital computation based on the use of interval numbers","author":"Moore, Ramon E.","year":"1965"},{"key":"12","volume-title":"Interval analysis","author":"Moore, Ramon E.","year":"1966"},{"key":"13","unstructured":"J.M. Muller, CRlibm. http:\/\/lipforge.ens-lyon.fr\/www\/crlibm\/, 2010."},{"key":"14","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1007\/BF01162949","article-title":"On the Phragm\u00e9n-Lindel\u00f6f theorem and some applications","volume":"72","author":"Rademacher, Hans","year":"1959","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"15","unstructured":"N. Revol and F. Rouillier, A library for arbitrary precision interval arithmetic, In 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, 2002."},{"issue":"203","key":"16","doi-asserted-by":"publisher","first-page":"415","DOI":"10.2307\/2152965","article-title":"Numerical computations concerning the ERH","volume":"61","author":"Rumely, Robert","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"276","key":"17","doi-asserted-by":"publisher","first-page":"2259","DOI":"10.1090\/S0025-5718-2011-02470-1","article-title":"Improvements to Turing\u2019s method","volume":"80","author":"Trudgian, Timothy","year":"2011","journal-title":"Math. 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(3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"19","series-title":"Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Fast Fourier transforms","author":"Walker, James S.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0849371546"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03077-X\/S0025-5718-2016-03077-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03077-X\/S0025-5718-2016-03077-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:57:06Z","timestamp":1776797826000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03077-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,15]]},"references-count":19,"journal-issue":{"issue":"302","published-print":{"date-parts":[[2016,11]]}},"alternative-id":["S0025-5718-2016-03077-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3077","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":[[2016,1,15]]}}}