{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:19:44Z","timestamp":1776845984405,"version":"3.51.2"},"reference-count":37,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,5,11]],"date-time":"2012-05-11T00:00:00Z","timestamp":1336694400000},"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 present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma Subscript k Baseline left-parenthesis a right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma _k(a)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Subscript n Superscript alpha\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">L_n^\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to investigate a certain sum\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript gamma Baseline left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S_\\gamma (n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    involving the constants\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma Subscript k Baseline left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma _k(1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript gamma Baseline left-parenthesis n right-parenthesis plus n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S_\\gamma (n)+n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which is consistent with the validity of the Riemann hypothesis.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02497-x","type":"journal-article","created":{"date-parts":[[2011,5,11]],"date-time":"2011-05-11T09:45:27Z","timestamp":1305107127000},"page":"2197-2217","source":"Crossref","is-referenced-by-count":12,"title":["An asymptotic form for the Stieltjes constants \ud835\udefe_{\ud835\udc58}(\ud835\udc4e) and for a sum \ud835\udc46\u1d67(\ud835\udc5b) appearing under the Li criterion"],"prefix":"10.1090","volume":"80","author":[{"given":"Charles","family":"Knessl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mark","family":"Coffey","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,5,11]]},"reference":[{"key":"1","unstructured":"M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Washington, National Bureau of Standards (1964)."},{"key":"2","first-page":"117","article-title":"Some constants associated with the Riemann zeta-function","volume":"3","author":"Briggs, William E.","year":"1955","journal-title":"Michigan Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-2285","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1006\/jnth.1999.2392","article-title":"Complements to Li\u2019s criterion for the Riemann hypothesis","volume":"77","author":"Bombieri, Enrico","year":"1999","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1016\/j.cam.2003.09.003","article-title":"Relations and positivity results for the derivatives of the Riemann \ud835\udf09 function","volume":"166","author":"Coffey, Mark W.","year":"2004","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s11040-005-7584-9","article-title":"Toward verification of the Riemann hypothesis: application of the Li criterion","volume":"8","author":"Coffey, Mark W.","year":"2005","journal-title":"Math. Phys. Anal. Geom.","ISSN":"https:\/\/id.crossref.org\/issn\/1385-0172","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"M. W. Coffey, Polygamma theory, the Li\/Keiper constants, and the Li criterion for the Riemann hypothesis, Rocky Mtn. J. Math. 40, 1841-1862, (2010).","DOI":"10.1216\/RMJ-2010-40-6-1841"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/j.jat.2006.10.006","article-title":"The theta-Laguerre calculus formulation of the Li\/Keiper constants","volume":"146","author":"Coffey, Mark W.","year":"2007","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1016\/j.jmaa.2005.06.048","article-title":"New results on the Stieltjes constants: asymptotic and exact evaluation","volume":"317","author":"Coffey, Mark W.","year":"2006","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"9","unstructured":"M. W. Coffey, Series representations for the Stieltjes constants, arXiv:0905.1111 (2009)."},{"key":"#cr-split#-10.1","unstructured":"M. W. Coffey, The Stieltjes constants, their relation to the  \u2c7c coefficients, and representation of the Hurwitz zeta function, arXiv\/math-ph\/0706.0343v2 (2007)"},{"key":"#cr-split#-10.2","doi-asserted-by":"crossref","unstructured":"Analysis 99, 1001-1021 (2010).","DOI":"10.1024\/1661-8157\/a000249"},{"key":"11","unstructured":"M. W. Coffey, Series of zeta values, the Stieltjes constants, and a sum \ud835\udc46\u1d67(\ud835\udc5b), arXiv\/math.ph\/0706.0345v2 (2007)."},{"key":"12","unstructured":"http:\/\/dlmf.nist.gov\/18.15"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"744","DOI":"10.1137\/0520052","article-title":"Uniform asymptotic expansions for Whittaker\u2019s confluent hypergeometric functions","volume":"20","author":"Dunster, T. M.","year":"1989","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"14","series-title":"Pure and Applied Mathematics, Vol. 58","volume-title":"Riemann's zeta function","author":"Edwards, H. M.","year":"1974"},{"issue":"5","key":"15","doi-asserted-by":"publisher","first-page":"1232","DOI":"10.1137\/0519087","article-title":"Uniform asymptotic expansions of Laguerre polynomials","volume":"19","author":"Frenzen, C. L.","year":"1988","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"16","isbn-type":"print","volume-title":"Table of integrals, series, and products","author":"Gradshteyn, I. S.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0122947606"},{"key":"17","unstructured":"G. H. Hardy, Note on Dr. Vacca\u2019s series for \ud835\udefe, Quart. J. Pure Appl. Math. 43, 215-216 (1912)."},{"key":"18","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"The Riemann zeta-function","author":"Ivi\u0107, Aleksandar","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/047180634X"},{"issue":"198","key":"19","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"},{"key":"20","unstructured":"J. C. Kluyver, On certain series of Mr. Hardy, Quart. J. Pure Appl. Math. 50, 185-192 (1927)."},{"issue":"273","key":"21","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1090\/S0025-5718-2010-02390-7","article-title":"An effective asymptotic formula for the Stieltjes constants","volume":"80","author":"Knessl, Charles","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"243","key":"22","doi-asserted-by":"publisher","first-page":"1379","DOI":"10.1090\/S0025-5718-02-01483-7","article-title":"Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants","volume":"72","author":"Kreminski, Rick","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1006\/jnth.1997.2137","article-title":"The positivity of a sequence of numbers and the Riemann hypothesis","volume":"65","author":"Li, Xian-Jin","year":"1997","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"49","DOI":"10.3836\/tjm\/1270133547","article-title":"On the power series coefficients of the Riemann zeta function","volume":"12","author":"Matsuoka, Yasushi","year":"1989","journal-title":"Tokyo J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0387-3870","issn-type":"print"},{"key":"25","isbn-type":"print","first-page":"279","article-title":"Generalized Euler constants associated with the Riemann zeta function","author":"Matsuoka, Y.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/9971978776"},{"key":"26","first-page":"395","article-title":"The signs of some constants associated with the Riemann zeta-function","volume":"9","author":"Mitrovi\u0107, Dragi\u0161a","year":"1962","journal-title":"Michigan Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-2285","issn-type":"print"},{"key":"27","doi-asserted-by":"publisher","first-page":"288","DOI":"10.2307\/2036349","article-title":"Asymptotic forms for Laguerre polynomials","volume":"24","author":"Muckenhoupt, Benjamin","year":"1970","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"28","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"},{"key":"29","unstructured":"B. Riemann, \u00dcber die Anzahl der Primzahlen unter einer gegebenen Gr\u00f6sse, Monats. Preuss. Akad. Wiss., 671 (1859-1860)."},{"key":"30","unstructured":"R. Smith, private communication (2010)."},{"key":"31","unstructured":"T. J. Stieltjes, Correspondance d\u2019Hermite et de Stieltjes, Volumes 1 and 2, Gauthier-Villars, Paris (1905)."},{"key":"32","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIII","volume-title":"Orthogonal polynomials","author":"Szeg\u0151, G\u00e1bor","year":"1975","edition":"4"},{"key":"33","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"},{"issue":"5","key":"34","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1137\/S0036141099359871","article-title":"Erratum to: \u201cWeighted zero distribution for polynomials orthogonal on an infinite interval\u201d [SIAM J. Math. Anal. 16 (1985), no. 6, 1317\u20131334; MR0807913 (87c:42024)]","volume":"32","author":"Van Assche, Walter","year":"2001","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"35","unstructured":"J. R. Wilton, A note on the coefficients in the expansion of \ud835\udf01(\ud835\udc60,\ud835\udc65) in powers of \ud835\udc60-1, Quart. J. Pure Appl. Math. 50, 329-332 (1927)."},{"issue":"2-3","key":"36","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1524\/anly.1994.14.23.147","article-title":"Some results on the generalized Stieltjes constants","volume":"14","author":"Zhang, Nan Yue","year":"1994","journal-title":"Analysis","ISSN":"https:\/\/id.crossref.org\/issn\/0174-4747","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02497-X\/S0025-5718-2011-02497-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02497-X\/S0025-5718-2011-02497-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:57:44Z","timestamp":1776790664000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02497-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,5,11]]},"references-count":37,"journal-issue":{"issue":"276","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["S0025-5718-2011-02497-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02497-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":[[2011,5,11]]}}}