627 Ramanujan 125 International Conference to Commemorate the 125th Anniversary of Ramanujan’s Birth Ramanujan 125 November 5–7, 2012 University of Florida, Gainesville, Florida Krishnaswami Alladi Frank Garvan Ae Ja Yee Editors AmericanMathematicalSociety Ramanujan 125 International Conference to Commemorate the 125th Anniversary of Ramanujan’s Birth Ramanujan 125 November 5–7, 2012 University of Florida, Gainesville, Florida Krishnaswami Alladi Frank Garvan Ae Ja Yee Editors Photograph of Participants, Ramanujan 125 Conference, Uni- versity of Florida, November 5–7, 2012. Courtesy of Ali Uncu. 627 Ramanujan 125 International Conference to Commemorate the 125th Anniversary of Ramanujan’s Birth Ramanujan 125 November 5–7, 2012 University of Florida, Gainesville, Florida Krishnaswami Alladi Frank Garvan Ae Ja Yee Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash C. Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 05A19, 11A25, 11E25, 11F33, 11F37, 11P84, 14K25, 17B67, 30B70, 33D15. Library of Congress Cataloging-in-Publication Data Ramanujun125: internationalconferencetocommemoratethe125thanniversaryofRamanujan’s birth, November 5–7, 2012, University of Florida, Gainesville, Florida / Krishnaswami Alladi, FrankGarvan,AeJaYee,editors. pagescm. –(Contemporarymathematics;volume627) ConferenceinhonorofIndianmathematicianSrinivasaRamanujanAiyangar. Includesbibliographicalreferences. ISBN978-1-4704-1078-0(alk. paper) 1. Ramanujan Aiyangar, Srinivasa, 1887-1920. 2. Functions, Theta–Congresses. 3. Lie algebras–Congresses. I. Alladi, Krishnaswami, editor of compilation. II. Garvan, Frank (FrankG.),1955- III.Yee,AeJa,1971-editorofcompilation. QA345.R36 2014 512(cid:2).7–dc23 2014010726 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) DOI:http://dx.doi.org/10.1090/conm/627 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2014bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 191817161514 Contents Preface vii Hecke grids and congruences for weakly holomorphic modular forms Scott Ahlgren and Nickolas Andersen 1 Knots and q-series George E. Andrews 17 A partition inequality involving products of two q-Pochhammer symbols Alexander Berkovich and Keith Grizzell 25 Analogues of Koshliakov’s Formula Bruce C. Berndt, Sun Kim, and Alexandru Zaharescu 41 How to prove Ramanujan’s q-continued fractions Gaurav Bhatnagar 49 A nonsingular Z curve of genus 4 3 H. M. Farkas, J. Y. Kaminski, and E. Yakubov 69 Ramanujan’s radial limits Amanda Folsom, Ken Ono, and Robert C. Rhoades 91 An identity that may have changed the course of history Michael D. Hirschhorn 103 The major index generating function of standard Young tableaux of shapes of the form “staircase minus rectangle” C. Krattenthaler and M. J. Schlosser 111 Convergence of random continued fractions Lisa Lorentzen 123 (cid:2) Tensor product decomposition of sl(n) modules and identities Kailash C. Misra and Evan A. Wilson 131 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis Jean-Louis Nicolas and Jonathan Sondow 145 A quaternionic proof of the representation formulas of two quaternary quadratic forms Cherng-tiao Perng 157 v Preface Aninternationalconference“Ramanujan125”washeldNovember5–7,2012,in Gainesville, Florida. The conference, whichwas organized by Krishnaswami Alladi andFrankGarvanoftheUniversityofFlorida, andAeJaYeeofThePennsylvania State University, attracted 70 active research participants from around the world. TheconferencewassupportedbygrantsfromtheNationalScienceFoundationand theNationalSecurityAgencyandbyfundsfromThePennsylvaniaStateUniversity through the NSF grant of Ae Ja Yee. We are most grateful for this support that was crucial to the success of the conference. The conference featured ten plenary talks of one hour each by leaders in the world of Ramanujan’s mathematics, and 40 shorter presentations including sev- eral by graduate students. These lectures discussed significant progress in various branches of mathematics in the quarter century since Ramanujan’s centennial — progressdirectlyrelatedtoRamanujan’sworkortopicswhoseoriginscanbetraced to Ramanujan’s discoveries. This Contemporary Mathematics volume is the refer- eedproceedingsoftheconferenceandcontainsresearchandexpositorypapersbased ontalksdeliveredatthe conference. Allpapershave beenarrangedinalphabetical order of the first author’s last name. In his last letter to Hardy in January 1920, Ramanujan communicated his discoveryofthemockthetafunctions,whichmimicthethetafunctionsinthesense that their coefficients can be estimated to the same degree of precision as in the case of objects expressible in terms of theta functions. The mock theta functions are now considered to be among Ramanujan’s deepest contributions. Ramanujan hadobtainedasymptoticevaluationsofthesemockthetafunctions,andinhisletter hadobservedthatifcertainwell-behavedanalyticexpressionsweresubtractedfrom the mock theta functions, then the resulting error would be bounded. He also indicated bounds in certain instances. For many years the exact links between mock theta functions and modular forms were unknown, and this was one of many such tantalizing mysteries. In thelast decade, KenOno, KathrinBringmann, and their collaboratorshave connected mock theta functions to harmonic Maass forms, thereby providing the key to unlock this mystery by developing the ideas in a fundamental 2003 PhD thesis of Sander Zwegers that was written under the direction of Don Zagier in Bonn. On the opening day of this conference, Ono announced for the first time his recent work with Amanda Folsom and Robert Rhoades, in which they obtain a precise expression for the bounded error Ramanujan indicated. We are pleased thatthethreeauthorshavecontributedapaperonthistopictothisContemporary Mathematics volume. vii viii PREFACE Ramanujan’sdiscoveryofcongruencesforthepartitionfunctionstartledHardy whoneverexpecteddivisibilitypropertiesforobjectsdefinedbyanadditiveprocess. Over the last century, Ramanujan’s congruences have inspired the more general studyofcongruencepropertiessatisfiedbycoefficientsofmodularforms. Following theleadofHecke,OliverAtkinobtainednewtypesofsuchcongruences. Thepaper by Ahlgren and Andersen on Hecke grids and congruences for weakly holomor- phic modular forms provides certain infinite families of weakly holomorphic forms, and establishes stronger versions of various congruences conjecturedby Honda and Kaneko using the Atkin operator. Ramanujanisconsideredthealltimegreatestintheareaofcontinuedfractions. Inone of his letterstoHardy in 1913, Ramanujan provided astounding evaluations of what is now called the Rogers–Ramanujan continued fraction, and claimed a lovely modular identity connecting the values of this fraction at arguments x and x5. It was results like these that convinced Hardy and his Cambridge colleagues that Ramanujan was in the league of Euler and Jacobi. For this reason, Michael Hirschhorn notes that this modular identity for the Rogers–Ramanujan continued fraction changed the course of history, and in his paper provides a much simpler proof of this gem of Ramanujan. The semi-expository paper of Gaurav Bhatnagar provides an elegant unified approach to several continued fraction of Ramanujan, including the Rogers–Ramanujan continued fraction. To supplement these two lovely papers on continued fractions, Lisa Lorentzen, an authority in this area, has contributed a paper rigorously studying the convergence of random continued fractions. InconnectionwiththetwoRogers–Ramanujanidentities, thelateLeonEhren- preis had asked whether one could prove that the coefficients in the first identity are larger than those of the second by not relying on the series forms but by just considering the congruential product representation. This was settled recently by Kevin Kadell. Motivated by this, Alex Berkovich and Keith Grizzell have taken up asystematic study more generally of finite products, and in their paper provide intricate injective maps to establish such positivity results. George Andrews, the leader in the theory of partitions and q-series, and on Ramanujan’s work, has been responsible for opening up several new directions of investigation in these fields motivated by surprising new connections between par- titions, q-series, and other areas of mathematics. Garoufaldis, Le, and Zagier dis- covered some remarkable q-series identities in the course of studying certain knots. In his paper, Andrews has generalized and refined these identities by introducing freeparameters; by proving themusing q-series, he has explainedtheir placein the hierarchical structure of q-hypergeometric identities. DuringtheRamanujanCentennial, BillGosperhumorously remarkedthatRa- manujan reaches out from his grave and snatches away your best formulas. That is, Ramanujan had anticipated much important work by later authors. A good ex- ample is an important formula of Koshliakov of 1929 which can actually be found in Ramanujan’s Lost Notebook. The Koshliakov formula is a functional equation for a certain Bessel function and involves the divisor function as well. Just as the theta transformation formula is related to the Riemann zeta function ζ(s), Koshli- akov’s formula is equivalent to a functional equation for ζ2(s). Bruce Berndt, Sun Kim, and Alexander Zaharescu provide new analogues to Koshliakov’s formula by PREFACE ix considering character analogues of the divisor function. They indicate applications to the positivity of the values L(1,χ) of L-functions involving even characters χ. The study of Lie algebras has led to the discovery of new Rogers–Ramanujan type identities. Kailash Misra and Evan Wilson study tensor product decomposi- tions of certain modules arising in the investigation of infinite dimensional Kac– Moody algebras, and deduce some new partition identities as a consequence. Sim- ilarly, the study of theta constants has also led to the discovery of new partition identitiesbyFarkasandKrainthelastdecade. Here,HerschelFarkas,J.Y.Kamin- ski and E. Yakubov, lay the foundations of a theory of thetaconstant identities for certain non-singular curves, and indicate applications. Highly composite numbers were first studied by Ramanujan. In the course of that study, Ramanujan considered also the maximal order of the sum of divi- sors function, and noted that the Riemann Hypothesis implied a precise estimate for its maximal order. In recent years, Nicolas and Robin have investigated the results in an unpublished second manuscript of Ramanujan on highly composite numbers. Here Jean-Louis Nicolas and Jonathan Sondow describe the fascinating historysurroundinghighlycompositenumbersanddiscussimportantcontributions by Ramanujan and later authors. ThecelebratedtheoremofLagrangethateverypositiveintegerisasumoffour squares motivated Ramanujan to write down 51 examples of universal quadratic forms, namely those that represent all positive integers. It is also of interest to determine the number of representations and express these in terms of sums of well-known multiplicative functions. Cherng-tiao Perng studies two quaternary quadratic forms and their number of representations and proves his results using the algebra of quaternions. Finally, the paper of Christian Krattenthaler and Michael Schlosser deals with the study of the major index generating function of Young tableaux, and uses techniques from the theory of multi-dimensional q-hypergeometric series, as well as transformation formulas for elliptic hypergeometric series. Thus, the volume containsacollectionofpapersspanningabroadspectrumofmathematicsandrep- resentingareaswhereRamanujanhashad,andcontinuestohave,amajorinfluence. We appreciate both the support of Ed Dunne and Christine Thivierge of the AMS, and their interest in having these proceedings as a part of the AMS Con- temporary Mathematics series. We are also thankful to Ms. Margaret Somers for efficiently handling the needs of the conference participants, as well as all of the localarrangementsfortheconference. WethankAliUncufortheconferencegroup photo. Krishnaswami Alladi, Gainesville, Florida, Frank Garvan, Gainesville, Florida, and Ae Ja Yee, State College, Pennsylvania March, 2014