{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:53:55Z","timestamp":1776848035133,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"301","license":[{"start":{"date-parts":[[2016,12,1]],"date-time":"2016-12-01T00:00:00Z","timestamp":1480550400000},"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 give a direct interpretation of the validity of the Riemann hypothesis for all zeros with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal fraktur upper I left-parenthesis rho right-parenthesis element-of left-parenthesis 0 comma upper T right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u2111\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c1\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Im (\\rho )\\in (0,T]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in terms of the prime-counting function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by proving that Schoenfeld\u2019s explicit estimates for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the Chebyshov functions hold as long as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4.92 StartRoot x slash log left-parenthesis x right-parenthesis EndRoot less-than-or-equal-to upper T\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4.92<\/mml:mn>\n                            <mml:msqrt>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\/<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mo>\n                                \u2061\n                                \n                              <\/mml:mo>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:msqrt>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4.92\\sqrt {x\/\\log (x)} \\leq T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>\n                  <p>\n                    We also improve some of the existing bounds of Chebyshov type for the function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3060","type":"journal-article","created":{"date-parts":[[2015,4,8]],"date-time":"2015-04-08T11:03:26Z","timestamp":1428491006000},"page":"2483-2498","source":"Crossref","is-referenced-by-count":21,"title":["Estimating \ud835\udf0b(\ud835\udc65) and related functions under partial RH assumptions"],"prefix":"10.1090","volume":"85","author":[{"given":"Jan","family":"B\u00fcthe","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,12,1]]},"reference":[{"key":"1","series-title":"Encyclopedia of Mathematics and its 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Weil\u2019s explicit formula","volume":"323","author":"Barner, Klaus","year":"1981","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"148","key":"3","doi-asserted-by":"publisher","first-page":"1361","DOI":"10.2307\/2006473","article-title":"On the zeros of the Riemann zeta function in the critical strip","volume":"33","author":"Brent, Richard P.","year":"1979","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","unstructured":"[B{\\\"u}t] J. B\u00fcthe, An improved analytic method for calculating \ud835\udf0b(\ud835\udc65), arXiv:1410.7008."},{"key":"5","unstructured":"[B{\\\"u}t14] J. B\u00fcthe, Untersuchung der Primzahlz\u00e4hlfunktion und verwandter Funktionen, Ph.D. thesis, Bonn University, March 2015."},{"issue":"293","key":"6","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1090\/S0025-5718-2014-02886-X","article-title":"New bounds for \ud835\udf13(\ud835\udc65)","volume":"84","author":"Faber, Laura","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","unstructured":"[FKBJ] J. Franke, Th. Kleinjung, J. B\u00fcthe, and A. Jost, A practical analytic method for calculating \ud835\udf0b(\ud835\udc65), Math. Comp., to appear."},{"key":"8","unstructured":"[Gou04] X. Gourdon, The 10\u00b9\u00b3 first zeros of the Riemann Zeta function and zeros computation at very large height, \\url{http:\/\/numbers.computation.free.fr\/Constants\/Miscellaneous\/zetazeros1e13-1e24.pdf}, October 2004."},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1137\/0519027","article-title":"Bounds for the tails of sharp-cutoff filter kernels","volume":"19","author":"Logan, B. F.","year":"1988","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"10","series-title":"AKP Classics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1201\/9781439864548","volume-title":"Asymptotics and special functions","author":"Olver, Frank W. J.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/1568810695"},{"issue":"2","key":"11","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. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"293","key":"12","doi-asserted-by":"publisher","first-page":"1521","DOI":"10.1090\/S0025-5718-2014-02884-6","article-title":"Computing \ud835\udf0b(\ud835\udc65) analytically","volume":"84","author":"Platt, David J.","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"211","DOI":"10.2307\/2371291","article-title":"Explicit bounds for some functions of prime numbers","volume":"63","author":"Rosser, Barkley","year":"1941","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"134","key":"14","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/2005976","article-title":"Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc65) and \ud835\udf13(\ud835\udc65). II","volume":"30","author":"Schoenfeld, Lowell","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","unstructured":"[Stu36] C. Sturm, Memoire sur les \u00e9quation diff\u00e9rentielles lin\u00e9aire du second ordre, J. Math. Pure Appl. (1) 1 (1836), 106\u2013186."},{"key":"16","doi-asserted-by":"crossref","unstructured":"[vM95] H. von Mangoldt, Zu Riemanns Abhandlungen \u201cUeber die Anzahl der Primzahlen unter einer gegebenen Gr\u00f6sse\u201d, J. Reine Angew. Math. 114 (1895), 255\u2013305.","DOI":"10.1515\/crll.1895.114.255"},{"key":"17","volume-title":"A Treatise on the Theory of Bessel Functions","author":"Watson, G. 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