{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:52:18Z","timestamp":1776797538420,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"296","license":[{"start":{"date-parts":[[2016,4,9]],"date-time":"2016-04-09T00:00:00Z","timestamp":1460160000000},"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                    Hiary has presented an algorithm which allows us to evaluate the truncated theta function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma-summation Underscript k equals 0 Overscript n Endscripts exp left-parenthesis 2 pi normal i left-parenthesis z k plus tau k squared right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:mi>exp<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sum _{k=0}^n \\exp (2\\pi \\mathrm {i} (zk+\\tau k^2))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to within\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"plus-or-minus epsilon\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03f5\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pm \\epsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis ln left-parenthesis StartFraction n Over epsilon EndFraction right-parenthesis Superscript kappa Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>ln<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                              <mml:mfrac>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mi>\n                                  \u03f5\n                                  \n                                <\/mml:mi>\n                              <\/mml:mfrac>\n                            <\/mml:mstyle>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03ba\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(\\ln (\\tfrac {n}{\\epsilon })^{\\kappa })<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    arithmetic operations for any real\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z\">\n                        <mml:semantics>\n                          <mml:mi>z<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\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                    . This remarkable result has many applications in Number Theory, in particular, it is the crucial element in Hiary\u2019s algorithm for computing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta left-parenthesis one half plus normal i t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                              <mml:mfrac>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mfrac>\n                            <\/mml:mstyle>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta (\\tfrac {1}{2}+\\mathrm {i} t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to within\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"plus-or-minus t Superscript negative lamda\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pm t^{-\\lambda }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O Subscript lamda Baseline left-parenthesis t Superscript one third Baseline ln left-parenthesis t right-parenthesis Superscript kappa Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>O<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mfrac>\n                                  <mml:mn>1<\/mml:mn>\n                                  <mml:mn>3<\/mml:mn>\n                                <\/mml:mfrac>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>ln<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03ba\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O_{\\lambda }(t^{\\frac {1}{3}}\\ln (t)^{\\kappa })<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    arithmetic operations. We present a significant simplification of Hiary\u2019s algorithm for evaluating the truncated theta function. Our method avoids the use of the Poisson summation formula, and substitutes it with an explicit identity involving the Mordell integral. This results in an algorithm which is efficient, conceptually simple and easy to implement.\n                  <\/p>","DOI":"10.1090\/mcom\/2953","type":"journal-article","created":{"date-parts":[[2015,4,9]],"date-time":"2015-04-09T10:52:05Z","timestamp":1428576725000},"page":"2911-2926","source":"Crossref","is-referenced-by-count":3,"title":["Computing the truncated theta function via Mordell integral"],"prefix":"10.1090","volume":"84","author":[{"given":"A.","family":"Kuznetsov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,4,9]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"Table of integrals, series, and products","author":"Gradshteyn, I. S.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780123736376","edition":"7"},{"issue":"2","key":"2","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. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"859","DOI":"10.4007\/annals.2011.174.2.3","article-title":"A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals","volume":"174","author":"Hiary, Ghaith Ayesh","year":"2011","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"539","DOI":"10.2307\/2005181","article-title":"A note on the evaluation of the complementary error function","volume":"26","author":"Hunter, D. B.","year":"1972","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-4","key":"5","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s11075-008-9191-x","article-title":"Some new applications of truncated Gauss-Laguerre quadrature formulas","volume":"49","author":"Mastroianni, G.","year":"2008","journal-title":"Numer. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/1017-1398","issn-type":"print"},{"key":"6","unstructured":"L. J. Mordell, The value of the definite integral \u222b_{-\u221e}^{\u221e}\\frac{\ud835\udc52^{\ud835\udc4e\ud835\udc61\u00b2+\ud835\udc4f\ud835\udc61}}\ud835\udc52^{\ud835\udc50\ud835\udc61}+\ud835\udc51\ud835\udc51\ud835\udc61, Quarterly Journal of Math. 68 (1920), 329\u2013342."},{"key":"7","doi-asserted-by":"crossref","unstructured":"L. J. Mordell, The definite integral \u222b_{-\u221e}^{\u221e}\\frac{\ud835\udc52^{\ud835\udc4e\ud835\udc61\u00b2+\ud835\udc4f\ud835\udc61}}\ud835\udc52^{\ud835\udc50\ud835\udc61}+\ud835\udc51\ud835\udc51\ud835\udc61 and the analytic theory of numbers, Acta Math. 61 (1933), 322\u2013360.","DOI":"10.1007\/BF02547795"},{"key":"8","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1007\/BF01436524","article-title":"Chebyshev expansions for Fresnel integrals","volume":"7","author":"N\u00e9meth, G.","year":"1965","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"9","unstructured":"S. Ramanujan, Some definite integrals connected with Gauss sums, Messenger of Mathematics 44 (1915), 75\u201385."},{"key":"10","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1137\/0709033","article-title":"Peano error estimates for Gauss-Laguerre quadrature formulas","volume":"9","author":"Stroud, A. H.","year":"1972","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"11","isbn-type":"print","volume-title":"The theory of the Riemann zeta-function","author":"Titchmarsh, E. C.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0198533691","edition":"2"},{"key":"12","doi-asserted-by":"crossref","unstructured":"W. Van Snyder, Algorithm 723: Fresnel integrals, ACM Trans. Math. Softw. 19 (1993), no. 4, 452\u2013456.","DOI":"10.1145\/168173.168193"},{"key":"13","unstructured":"S. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University, arXiv:0807.4834, 2002."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-296\/S0025-5718-2015-02953-6\/S0025-5718-2015-02953-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-296\/S0025-5718-2015-02953-6\/S0025-5718-2015-02953-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:33:22Z","timestamp":1776796402000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-296\/S0025-5718-2015-02953-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,9]]},"references-count":13,"journal-issue":{"issue":"296","published-print":{"date-parts":[[2015,11]]}},"alternative-id":["S0025-5718-2015-02953-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/2953","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":[[2015,4,9]]}}}