{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:36:17Z","timestamp":1776800177981,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"298","license":[{"start":{"date-parts":[[2016,7,22]],"date-time":"2016-07-22T00:00:00Z","timestamp":1469145600000},"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                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\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\">p(n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote the partition function. DeSalvo and Pak proved that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction p left-parenthesis n minus 1 right-parenthesis Over p left-parenthesis n right-parenthesis EndFraction left-parenthesis 1 plus StartFraction 1 Over n EndFraction right-parenthesis greater-than StartFraction p left-parenthesis n right-parenthesis Over p left-parenthesis n plus 1 right-parenthesis EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\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:mfrac>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mfrac>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mfrac>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\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:mrow>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {p(n-1)}{p(n)}\\left (1+\\frac {1}{n}\\right )&gt; \\frac {p(n)}{p(n+1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Moreover, they conjectured that a sharper inequality\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction p left-parenthesis n minus 1 right-parenthesis Over p left-parenthesis n right-parenthesis EndFraction left-parenthesis 1 plus StartFraction pi Over StartRoot 24 EndRoot n Superscript 3 slash 2 Baseline EndFraction right-parenthesis greater-than StartFraction p left-parenthesis n right-parenthesis Over p left-parenthesis n plus 1 right-parenthesis EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\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:mfrac>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mfrac>\n                                <mml:mi>\n                                  \u03c0\n                                  \n                                <\/mml:mi>\n                                <mml:mrow>\n                                  <mml:msqrt>\n                                    <mml:mn>24<\/mml:mn>\n                                  <\/mml:msqrt>\n                                  <mml:msup>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mn>3<\/mml:mn>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mo>\/<\/mml:mo>\n                                      <\/mml:mrow>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:mrow>\n                                  <\/mml:msup>\n                                <\/mml:mrow>\n                              <\/mml:mfrac>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\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:mrow>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {p(n-1)}{p(n)}\\left ( 1+\\frac {\\pi }{\\sqrt {24}n^{3\/2}}\\right ) &gt; \\frac {p(n)}{p(n+1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    holds for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to 45\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>45<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq 45<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"minus normal upper Delta squared log p left-parenthesis n minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/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\">-\\Delta ^{2} \\log p(n-1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u0394\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the difference operator with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We also show that for given\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r\\geq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and sufficiently large\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis negative 1 right-parenthesis Superscript r minus 1 Baseline normal upper Delta Superscript r Baseline log p left-parenthesis n right-parenthesis greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>r<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>r<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(-1)^{r-1}\\Delta ^{r} \\log p(n)&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r\\geq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta Superscript r Baseline p left-parenthesis n right-parenthesis greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>r<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta ^{r} p(n)&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for sufficiently large\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/2999","type":"journal-article","created":{"date-parts":[[2015,7,22]],"date-time":"2015-07-22T09:00:00Z","timestamp":1437555600000},"page":"825-847","source":"Crossref","is-referenced-by-count":28,"title":["Finite differences of the logarithm of the partition function"],"prefix":"10.1090","volume":"85","author":[{"given":"William","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Larry","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gary","family":"Xie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,7,22]]},"reference":[{"key":"1","unstructured":"M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York 1965."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/0022-314X(91)90080-U","article-title":"Exact asymptotic formulas for the coefficients of nonmodular functions","volume":"38","author":"Almkvist, Gert","year":"1991","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"173","DOI":"10.4064\/aa-61-2-173-181","article-title":"On the differences of the partition function","volume":"61","author":"Almkvist, Gert","year":"1992","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1017\/S0013091508001016","article-title":"Bounds for modified Bessel functions of the first and second kinds","volume":"53","author":"Baricz, \u00c1rp\u00e1d","year":"2010","journal-title":"Proc. Edinb. Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-0915","issn-type":"print"},{"key":"5","unstructured":"C. Bessenrodt and K. Ono, Maximal multiplicative properties of partitions, Ann. Combin. to appear."},{"key":"6","unstructured":"W. Y. C. Chen, Recent developments on log-concavity and \ud835\udc5e-log-concavity of combinatorial polynomials, DMTCS Proceeding, 22nd International Conference on Formal Power Series and Algebraic Combinatorics, 2010."},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/BF02124750","article-title":"On the Lambert \ud835\udc4a function","volume":"5","author":"Corless, R. M.","year":"1996","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"S. DeSalvo and I. Pak, Log-concavity of the partition function, The Ramanujan Journal, to appear, DOI:10.1007\/s11139-014-9599-y.","DOI":"10.1007\/s11139-014-9599-y"},{"key":"9","unstructured":"I. J. Good, The difference of the partition function, Problem 6137, Amer. Math. Monthly 84 (1997), 141."},{"issue":"144","key":"10","doi-asserted-by":"publisher","first-page":"1241","DOI":"10.2307\/2006348","article-title":"Finite differences of the partition function","volume":"32","author":"Gupta, Hansraj","year":"1978","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","isbn-type":"print","first-page":"276","article-title":"Asymptotic formul\u00e6in combinatory analysis [Proc. London Math. Soc. (2) 17 (1918), 75\u2013115]","author":"Hardy, G. H.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0821820761"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1137\/0150020","article-title":"Partition asymptotics from recursion equations","volume":"50","author":"Knessl, Charles","year":"1990","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"issue":"8-9","key":"13","doi-asserted-by":"publisher","first-page":"1033","DOI":"10.1002\/cpa.3160440814","article-title":"Asymptotic behavior of high-order differences of the partition function","volume":"44","author":"Knessl, Charles","year":"1991","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"271","DOI":"10.2307\/1990043","article-title":"On the series for the partition function","volume":"43","author":"Lehmer, D. H.","year":"1938","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"362","DOI":"10.2307\/1989927","article-title":"On the remainders and convergence of the series for the partition function","volume":"46","author":"Lehmer, D. H.","year":"1939","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"3","key":"16","doi-asserted-by":"publisher","first-page":"237","DOI":"10.4064\/aa-49-3-237-254","article-title":"Differences of the partition function","volume":"49","author":"Odlyzko, A. M.","year":"1988","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1112\/S0025579300012602","article-title":"Asymptotic expansions and inequalities for hypergeometric functions","volume":"44","author":"Ponnusamy, S.","year":"1997","journal-title":"Mathematika","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5793","issn-type":"print"},{"key":"18","doi-asserted-by":"crossref","unstructured":"H. Rademacher, A convergent series for the partition function \ud835\udc5d(\ud835\udc5b), Proc. Nat. Acad. Sci. 23 (1937), 78\u201384.","DOI":"10.1073\/pnas.23.2.78"},{"key":"19","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 169","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-80615-5","volume-title":"Topics in analytic number theory","author":"Rademacher, Hans","year":"1973"},{"key":"20","volume-title":"A Treatise on the Theory of Bessel Functions","author":"Watson, G. N.","year":"1944"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-02999-8\/S0025-5718-2015-02999-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-02999-8\/S0025-5718-2015-02999-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:42:15Z","timestamp":1776796935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-298\/S0025-5718-2015-02999-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,22]]},"references-count":20,"journal-issue":{"issue":"298","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["S0025-5718-2015-02999-8"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/2999","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":[[2015,7,22]]}}}