Springer Proceedings in Mathematics & Statistics

George E. AndrewsFrank Garvan Editors

Analytic Number Theory, Modular Forms and q-Hypergeometric SeriesIn Honor of Krishna Alladi’s 60th Birthday, University of Florida, Gainesville, March 2016

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George E. Andrews • Frank GarvanEditors

Analytic Number Theory,Modular Formsand q-Hypergeometric SeriesIn Honor of Krishna Alladi’s 60th Birthday,University of Florida, Gainesville, March2016

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EditorsGeorge E. AndrewsDepartment of MathematicsPennsylvania State UniversityState College, PAUSA

Frank GarvanDepartment of MathematicsUniversity of FloridaGainesville, FLUSA

ISSN 2194-1009 ISSN 2194-1017 (electronic)Springer Proceedings in Mathematics & StatisticsISBN 978-3-319-68375-1 ISBN 978-3-319-68376-8 (eBook)https://doi.org/10.1007/978-3-319-68376-8

Library of Congress Control Number: 2017958728

Mathematics Subject Classification (2010): 05A17, 11B65, 11F30, 11K65, 11N05, 11P84, 33D15

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Preface

We are extremely pleased to offer these proceedings of the Gainesville NumberTheory Conference of 2016, more affectionately known as ALLADI60, honoringKrishna Alladi on his 60th birthday. Krishna has been a major contributor tonumber theory and mathematics in several ways. First, he is a first-class mathe-matician. We who have collaborated with him are most vividly aware of his insightand talent. Second, he is Editor-in-chief of the Ramanujan Journal and a SeriesEditor of Developments in Mathematics. Third, he instituted the internationallyadmired SASTRA Ramanujan Prize. Fourth, he has been the major actor in thecreation of numerous important conferences.

In the past two decades, the University of Florida has been the main internationalvenue for conferences in the areas of partitions, q-series, modular forms, andRamanujan’s work. The special feature of this conference was that in addition tothese areas, analytic number theory, irrationality, and transcendence were alsocovered—areas that Alladi had worked on until 1990.

The conference attracted nearly 200 participants and was spread over 5 days toaccommodate the nearly 100 speakers who had come from Australia, Austria,Canada, China, England, France, Germany, Hong Kong, Hungary, India, Israel,Korea, The Netherlands, New Zealand, Norway, Serbia, Switzerland, Tunisia,Turkey, and the USA. The conference was supported by grants from the NationalScience Foundation, the National Security Agency, the Number TheoryFoundation, and by funds from The Pennsylvania State University. Local supportwas provided by the University of Florida, Mathematics Department and theCollege of Liberal Arts and Sciences, and the Alachua County Tourism Board.We are most grateful for all this support which helped make the conference asuccess.

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Special Lectures

There were four special lectures at the conference:

Opening LectureManjul Bhargava (Princeton University)Squarefree values of polynomial discriminantsIn 2003, Manjul Bhargava won the First SASTRA Ramanujan Prize, a prize thatKrishna Alladi was instrumental in launching.

The Erdős ColloquiumHugh L. Montgomery (University of Michigan)Littlewood polynomialsThis Special Colloquium, initiated by Krishna Alladi, has been given yearly at theUniversity of Florida since 1999. Following the Colloquium, the participants weretreated to a dinner party at the home of Krishna and Mathura Alladi, and our thanksto Mathura for graciously hosting this.

The Ramanujan ColloquiumJames Maynard (University of Oxford)Linear equations in primesThis Special Colloquium, initiated by Krishna Alladi, has been given yearly at theUniversity of Florida since 2007, and is sponsored by The Pennsylvania StateUniversity and George Andrews. James Maynard also gave two other lectures.

Conference Closing Lecture and Math ColloquiumWadim Zudilin (University of Newcastle)Short random walks and Mahler measuresWadim Zudilin also gave a lecture on certain irrational values of the logarithm,which was related to Krishna Alladi’s work from 1979.

Other Conference Highlights

Piano RecitalFollowing Maynard’s Colloquium, there was a Reception at the Keene FacultyCenter where the participants were treated to a lovely piano concert by ChristianKrattenthaler of the University of Vienna.

AwardsRon Graham, former President of the AMS, presented cheques to James Maynardand Kevin Ford in recognition of their work for the resolution of a famous $10,000problem of Erdős on large gaps between primes. Maynard received $5,000 for hissolo paper, and Kevin Ford received a cheque for $5,000 made out to Kevin Ford,Ben Green, S. Konyagin, and Terence Tao for their joint work.

vi Preface

Soundararajan’s LectureAnother SASTRA Ramanujan prize winner, Kannan Soundararajan (Stanford),presented for the first time his recent work with his colleague Robert Lemke Oliveraon a startling new result on a bias in the distribution of consecutive prime numbers.

The Man Who Knew InfinityThe movie The Man Who Knew Infinity on the remarkable life of Ramanujan wasshown at the conference as a Preview before its official opening in theaters thanks tothe efforts of Manjul Bhargava, one of the Associate Producers of the movie. Ourthanks to Edward Pressman films for this kind gesture. See Bruce Berndt’s paper inthis volume for some background on an interesting scene from the movie.

Talks on Alladi’s WorkSeveral talks at the conference dealt with Alladi’s work—not just his recent workon partitions and q-series, but his early work as well. Dorian Goldfeld (ColumbiaUniversity) spoke about extensions of results in Alladi’s first paper (written whenhe was an undergraduate) with Paul Erdős on an additive arithmetic function—nowcalled the Alladi-Erdős function. Doron Zeilberger (Rutgers University) and WadimZudilin (University of Newcastle, Australia) discussed extensions of results ofAlladi-Robinson (1980) on irrationality measures.

Conference BanquetA Conference Banquet was held at the Paramount Hotel. Many speeches honoringKrishna were given. The text of Elizabeth Loew’s speech is given after this preface.

Mathematical Interests of Krishna Alladi

Over his career, Krishna Alladi has maintained an interest in Number Theory,Combinatorics, Discrete Mathematics, Analytic Number Theory, Sieve Methods,Probabilistic Number Theory, Diophantine Approximations, Partitions, andq-Series Identities, His research in mathematics began as an 18-year-old under-graduate in 1973. The first part of his mathematical career was in Analytic NumberTheory. In particular, he wrote five joint papers with Paul Erdős. In 1987, theRamanujan Centenary year, Krishna became interested in partitions and in the early1990s, he began a fruitful collaboration with George Andrews, Alex Berkovich, andBasil Gordon, when they made impressive breakthroughs in discovering partitionidentities beyond those of Rogers-Ramanujan, Schur and Göllnitz. In 1993, hespent a sabbatical at Penn State with George Andrews. There he learned theimportance of basic hypergeometric series, and modular forms for the theory ofpartitions.

Preface vii

Volume Contents

Below, we give a brief description of the papers in this volume and group themaccording to these topics: Analytic Number Theory, Probabilistic Number Theory,Partitions, Basic Hypergeometric Functions, and Modular Forms.

Analytic Number TheoryBenli, Elma, and Yidirim extend Conrey and Ghosh’s results on zeros of derivativesof the Riemann zeta-function near the critical line, to Dirichlet L-functions.

Deshouillers and Grekos study the problem of the number of integral points on aconvex curve in terms of length and curvature, and make improvements on previousresults.

In 1977, Alladi and Erdős showed that a certain important additive function isuniformly distributed modulo 2. Goldfeld generalizes this result to an arbitrarymodulus.

Montgomery’s survey paper on Littlewood Polynomials is an expanded versionof his talk given at the conference.

Nicolas obtains an effective version of Ramanujan’s result for the differencebetween the logarithmic integral of Chebychev’s function and pðxÞ.

Ono, Schneider, and Wagner prove a partition theoretic analog of Alladi’sMöbius function identity.

Saradha and Sharma prove some conjectures of Mueller and Schmidt for thenumber of integer solutions of a Thue inequality for certain binary quadratic forms.

Tenenbaum extends a result of Mertens for the sum of the reciprocals of primesto the sum of the reciprocals of the product of k primes.

Inspired by Hugh Montgomery’s talk on Littlewood polynomial, Zeilberger(with his computer collaborator Ekhad) gives an algorithmic approach to Saffari’sconjecture on the asymptotic growth of moments of the Rudin–Shapiropolynomials.

Probabilistic Number TheoryElliott’s paper is survey of abstract multiplicative functions and their application tothe study of the Fourier coefficients of automorphic forms, together with a dis-cussion in the context of the theory of Probabilistic Number Theory.

PartitionsGeorge Andrews recently gave a refinement of Krishna Alladi’s variant of Schur’s1926 partition theorem. In his paper, Andrews develops a surprising factorizationof the related polynomial generating functions.

Chen, Ji, and Zang previously proved a rank-crank inequality conjecture ofAndrews, Dyson, and Rhoades. By using combinatorial methods, they show thatthere is a reordering of partitions that explains the very nearly equal distributionsof the rank and the crank.

Motivated by recent research of Krishna Alladi, Berkovich, and Uncu give newweighted partition identities for partitions, overpartitions, and partitions with dis-tinct even parts, using the theory of basic hypergeometric functions.

viii Preface

Dousse extends Krishna Alladi’s method of weighted words to obtain general-izations and refinements for previous extensions of Schur’s partition theorem tooverpartitions due to Andrews, Corteel, and Lovejoy.

Kolitsch gives new partition interpretations of truncated forms of Euler’sPentagonal Number Theorem and Jacobi’s Triple Product identity in terms ofoverpartitions.

Krattenthaler finds congruences mod 16 for the number of unique path partitionsof n, which occur in the study of character values of finite symmetric groups, andwhich generalize results of Olsson, Bessendrodt, and Sellers.

Kanade, Kurşungöz, and Russel give combinatorial interpretations of overpar-tition variants of Andrews’s H and J functions which occurred in the study of theAndrews-Gordon partition identities and their generalizations.

Lovejoy gives two overpartition extensions of Alladi and Gordons generaliza-tion of Schurs theorem.

Bringmann and Mahlburg present new companions to the Capparelli partitionidentities and two new general identities for three-color partitions that may bespecialized to theta functions and false theta functions.

Seo and Yee give a combinatorial proof of a result of Andrews which is anoverpartition analog of Rogers-Ramanujan type theorem related to restricted suc-cessive ranks.

q-Series and Basic Hypergeometric FunctionsGaurav Bhatnagar gives a marvelous bibasic version of Heine’s basic hypergeo-metric transformation and uses it to prove and organize a raft of identities ofRamanujan, some of which are easy and some which are not.

Cooper, Wan, and Zudilin prove a number of Z.-W. Sun’s conjectures for seriesfor 1=p by relating them to known series using techniques of basic hypergeometricseries and Zeilberger’s algorithm for holonomic sequences.

Banerjee and Dixit obtain new identities for Ramanujan’s function rðqÞ which isthe generating function for the excess number of partitions of n into distinct partswith even rank over those of odd rank.

Hirschhorn gives elementary proofs of some well-known arithmetic properties ofRamanujan’s tau function using nothing more than Jacobi’s triple product identity.

Liu describes a method for finding certain series expansions of functions thatsatisfy a q-partial differential equation and, as an application, finds a generalizationof Andrews’s transformation formula for the q-Lauricella function.

Mc Laughlin gives a new approach using bilateral hypergeometric series toobtain identities for mock theta functions and finds radial limit formulas as anapplication.

Schlosser and Yoo employ a one-variable extension of q-rook theory to givecombinatorial proofs of some basic hypergeometric summations formulas.

Sills gives an elementary approach for finding the sum side of Rogers-Ramanujan type identities from the product forms related to the standard modules

of the Kac–Moody algebra Að2Þ2 .

Preface ix

Modular FormsNicolas Andersen follows up on his previous work, “Vector-Valued ModularForms and The Mock Theta Conjectures,” where he gave a new proof ofRamanujan’s fifth-order mock theta conjectures using the theory of vector-valuedmodular forms and harmonic Maass forms. In his new paper, he extends these ideasto give a new proof of Hickerson’s seventh-order identities.

Jha and Kumar compute the adjoint (with respect to the Petersson inner product)of the linear map related to the Cohen–Rankin bracket, thus extending work ofKohnen and Herrero to half-integral weight modular forms.

Kimport obtains asymptotic expansions for weight 1=2 and 3=2 partial thetafunctions at roots of unity which generalizes results of Berndt and Kim and areimportant in the study of certain quantum modular forms.

McIntosh proves that Zweger’s l-function, which is important in the study ofmock theta functions from a modular form view, is essentially no more general thanthe universal mock theta function g2.

Paule and Radu derive a new type of modular function identity that impliesRamanujan’s partition congruence mod 11.

Ramakrishnan, Sahu, and Singh use the theory of modular forms to find for-mulas for the number of representations of a positive integer by certain class ofquadratic forms in eight variables.

Jon Borwein

Jon Borwein was unable to come to ALLADI60 due to his commitment to give aseries of lectures as a Distinguished Scholar in Residence at Western University,London Ontario. Later, in May 2016, Jon and his wife Judi were able to visit us inFlorida. He gave two talks—one at the University of Florida Brain Institute onCARMA: A Model for Multi-Discipline and Multi-Institution Collaboration. Theother talk on Seeing Things by Walking on Numbers was given in the MathDepartment. Later, in July, I (Frank) was with Jon at the Lambert Conference inLondon, Ontario. It was a great shock to us that he died just a few days after theLambert Conference, and we still feel the loss. Jon was the first person to submit apaper to our proceedings. The referee only required some minor revisions. Jonwanted to wait until he got back to Australia to complete these revisions but sadlythis did not happen. I made the revisions myself and got David Bailey to checkthem over. Jon’s paper is an expanded companion to a talk he gave at a workshopcelebrating Tony Guttmann’s 70th birthday. It describes his encounters over nearly30 years with Sloane’s (Online) Encyclopedia of Integer Sequences. We agree withthe referee that it is a masterpiece, with beautiful math and beautiful exposition.

x Preface

Alladi Ramakrishnan

Krishna’s father, the late Prof. Alladi Ramakrishnan, the Founder-Director ofMATSCIENCE, The Institute of Mathematical Sciences in Madras, India, was aninspiration to Krishna and supported all of Krishna’s efforts. In an emotionallycharged speech at the banquet, Krishna said that if his father were alive, he wouldhave been the happiest person to see such an impressive gathering of mathemati-cians from around the world for the 60th birthday conference. Several speakers intheir speeches at the banquet made references to Krishna’s father. Kryuchkov,Lanfear, and Suslov have dedicated their paper to the memory of Krishna’s father,the famous physicist Prof. Alladi Ramakrishnan, on the topic of a complex form ofclassical and quantum electrodynamics.

Thanks

We thank Marc Strauss and Elizabeth Loew for the Springer book exhibit andRochelle Kronzek for the World Scientific book exhibit. In addition, we thank MarcStrauss for publishing these proceedings in the Springer Proceedings Series. Weexpress special thanks to Margaret Somers, Cyndi Garvan, Ali Uncu, and ChrisJennings-Shaffer, the staff of the Math Department and the number theory graduatestudents for all aspects of preparing for and running a smooth conference. We thankJis Joseph for his photography and help with other aspects of the conference.

We conclude by expressing again our thanks to Krishna for his monumentalcontributions. Our community has been greatly enriched by him, and we are deeplyin his debt.

Gainesville, FL, USA George E. AndrewsAugust 2017 Frank Garvan

Preface xi

Introductory Speech

Speech Given by Elizabeth Loew at the ALLADI60 Banquet

I am especially honored to have been asked by George Andrews to give a shortspeech tonight. Thank you George. I also want to thank Frank Garvan for invitingme to share in this important celebration in honor of Krishna’s 60th birthday.

***While I don’t quite remember the first time that I met Krishna, it must have beensometime in 2008, here at UFL, after I had become a senior editor with Springer.Our meeting was certainly during Krishna’s tenure as Chair of the department.Among my responsibilities for journals and books were The Ramanujan Journaland the series Developments in Mathematics, both of which were founded byKrishna. Both became demanding, because they were intellectually rich and Iwanted to ensure that I was doing my part to reinvigorate and help expand frompublishing capabilities. This was enjoyable for me personally.

***It was politely suggested to me that we weren’t coming here tonight to roastKrishna. (I assume that everyone knows what I mean by ROAST). That is, I am notgoing to say anything particularly funny or amusing. The Krishna that I have cometo know is a serious kind of guy. But I have to say, that in the midst of situationsand discussions, he has often made me laugh. He has certain unique expressionsand ways of politely conveying dissatisfaction. From my perspective, however,Krishna has always been above all forms of pettiness. Over the years, I have greatlyappreciated his warmth and genuine friendship to me. This extends to his lovelywife, Mathura, and all of the Alladi children. I have been privileged to enjoy manyevenings in their home and I wish the Alladi family many years of continuedhappiness and good health.

***Krishna and I have worked closely and well together for these last 8 years and Iwould like to comment and recollect on a few important things about the publi-cations we continue to work on together. Krishna is both the founder and

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Editor-in-chief of The Ramanujan Journal. Frank Garvan is the journal’s extremelycapable managing editor. And many of you here today are active members of theboard. The journal was established in 1997 with Kluwer and upon the Kluwer andSpringer merger, RAMA became a Springer publication. There is no need toconvey to everyone here the importance of Ramanujan himself and the enormousimpact his genius had and still has on the mathematics community. So RAMA isdevoted to those aspects of mathematics influenced by Ramanujan. Indeed duringmy years at Springer, these influences have grown substantially. The journal hadalways been strong in covering q-series and special functions, for example, and inthe last few years modular forms has become a critical area of coverage for thejournal as well. Various topics have expanded greatly, some have even ‘exploded’(to use Krishna’s wording). I just want to emphasize that the journal continues toreinforce connections and to establish links with new fields. And I would like to putit this way: the editorial board is choc full of valued mathematicians all of whomare experts in their areas of research. The journal publishes 9 issues per year.Special issues are published from time to time with respect to the Sastra Ramanujanlectures, covering the 125th anniversary of Ramanujan, memorial issues for suchgreat figures as Erdős, Rankin, Gordon, and a forthcoming one for MarvinKnopp. Special issues have also been published for milestone birthdays such as forRichard Askey and George Andrews, to name just two. So, on behalf of Springer Iwould like to thank Krishna, Frank, and the entire board for their continueddevotion to enhancing the quality and excellence of RAMA.

***The book series Developments in Mathematics was founded by Krishna as pre-viously mentioned. When I entered the picture Bob Guralnick was co-series editorand eventually stepped down. In 2009 Hershel Farkas joined as co-series editor.The 3 of us have been working effectively and diligently toward expanding theseries. This too has been an enriching collaboration for me. In earlier years, mostof the publications in DEVM included edited works, most often in the area ofnumber theory, but in other areas as well. Nowadays, DEVM predominantlyintends to publish quality monographs. The aim being to cover new topics in theforefront of mathematical research; topics that are perhaps a bit more specialized.While new topics in number theory have published in DEVM, new books in theseries include those in fields such as differential equations, algebra, enumerativecombinatorics, analysis, to name several. Quality edited volumes are still occa-sionally considered for publication. A selection of edited volumes that may be ofinterest to you are displayed here at the conference “Partitions, q-Series, andModular Forms” (Alladi-Garvan), “Quadratic and Higher Degree Forms”(Alladi, Barghava, Savitt, Tiep), “Surveys in Number Theory” (Alladi), and“Combinatory Analysis” (Alladi, Paule, Sellers, Yee).

***A word about Krishna’s father. I remember Krishna’s father very well. My parents,Bert and Ann Kostant, also had the pleasure of meeting Alladi Ramakrishnan. Withhis kind face and beautiful smile, he was a very interesting person. Discussions withhim took place in Krishna’s home where I recall (and as Bert also told me) hearing a

xiv Introductory Speech

variety of thoughts and recollections that came to his father’s mind. When Dr.Ramakrishnan passed away, I remember remarking to Krishna what a devoted sonhe had been to his father. The magnificent book that Krishna put together entitledThe Legacy of Alladi Ramakrishnan in the Mathematical Sciences is on displayhere and is a beautiful intellectual and sentimental tribute to a highly respectedmathematical physicist. I felt happy to be involved in publishing that Work.

***The book Ramanujan’s Place in the World of Mathematics was published in2013 with my colleagues in Springer India. This book represents a compendium ofKrishna’s articles over time and has received great reviews from various sources. Init, the reader learns of Ramanujan and his mathematical work in an historicalcontext and with comparisons to other great mathematicians throughout history.I hope that there are several copies of this book here at the conference.

***Krishna is a creator of original mathematical research; a composer of books; abuilder of two noteworthy publishing venues, journal and book series. And againand again, over and over, I have been enriched and delighted to be part of much ofKrishna’s publishing activities. I consider myself rather fortunate to have cultivatedseveral strong and enriching relationships within the mathematics community.There is strong and enriching, and there is strong and enriching PLUS. I believe thatmy relationship with Krishna includes the PLUS. From Springer, from JoachimHeinze, from Marc Strauss, from myself, my son Max, and Ann and Bert Kostant,we wish Krishna healthy years ahead filled with as much joy as he can take in.

Introductory Speech xv

List of Participants

S. D. Adhikari, Harish-Chandra InstituteScott Ahlgren, University of IllinoisYildirim Akbal, Bilkent UniversityElie Alhajjar, George Mason University, VirginiaKrishnaswami Alladi, University of FloridaBadria Alsulmi, Kansas State UniversityNickolas Andersen, University of IllinoisGeorge Andrews, Pennsylvania State UniversityVictor Manuel Aricheta, Emory UniversityRichard Askey, University of WisconsinRoger Baker, Brigham Young UniversityWilliam Banks, University of MissouriOlivia Beckwith, Emory UniversityLea Beneish, Emory UniversityKübra Benli, University of GeorgiaAlexander Berkovich, University of FloridaBruce Berndt, University of IllinoisEdward Bertram, University of Hawaii at ManoaManjul Bhargava, Princeton UniversityGaurav Bhatnagar, University of ViennaMatt Boylan, University of South CarolinaKathrin Bringman, University of CologneDale Brownawell, Pennsylvania State UniversityHannah Burson, University of IllinoisNeil Calkin, Clemson UniversityZhu Cao, Kennesaw State UniversitySong Heng Chan, Nanyang Technological University, SingaporeDiego Chaves, Pennsylvania State UniversityYoun-seo Choi, KIASFatma Cicek, University of RochesterShaun Cooper, Massey University, New Zealand

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Hédi Daboussi, Université de Picardie, Amiens,FranceColin Defant, University of FloridaJean-Marc Deshouillers, Bordeaux INPHarold Diamond, University of IllinoisAtul Dixit, Indian Institute of Technology, GandhinagarJehanne Dousse, Universität ZürichDennis Eichhorn, University of California, IrvineAli-Bulent Ekin, Ankara UniversityPeter Elliott, University of Colorado, BoulderErtan Elma, University of WaterlooGeremias Polanco Encarnaci, Hampshire CollegeLarry Ericksen, Millville, New JerseyHershel Farkas, The Hebrew University of JerusalemEndrit Fejzullahu, University of FloridaAmanda Folsom, Amherst CollegeKevin Ford, University of IllinoisCraig Franze, The Ohio State University at MarionRichard Frnka, Louisiana State UniversityAyla Gafni, Pennsylvania State UniversityCyndi Garvan, University of FloridaFrank Garvan, University of FloridaMalay Ghosh, University of FloridaDorian Goldfeld, Columbia UniversityDan Goldston, San Jose State UniversityOscar Gonzalez, University of Puerto Rico at Rio PiedrasAnkush Goswami, University of FloridaRon Graham, University of California San DiegoMichael Griffin, Princeton UniversityPavel Guerzhoy, University of HawaiiMike Hirschhorn, University of New South Wales, AustraliaTim Huber, University of Texas Rio Grande ValleyMourad Ismail, University of Central FloridaAleksandar Ivic, Serbian Academy of SciencesMarie Jameson, University of TennesseeMin-Joo Jang, University of ColognePaul Jenkins, Brigham Young UniversityChris Jennings-Shaffer, Oregon State UniversityAbhas Kuma Jha, NISER, Bhubaneswar, IndiaJin Seokho, KIASJason Johnson, University of FloridaJunsoo Ha, KIASBen Kane, University of Hong KongK. Kannan, SASTRA UniversityKevin Keating, University of FloridaWilliam Keith, Michigan Technical University

xviii List of Participants

Byungchan Kim, Seoul National University of Science and TechnologySusie Kimport, Stanford UniversitySun Kim, University of IllinoisLouis Kolitsch, The University of Tennessee at MartinAnant Kota, University of California, BerkeleyChristian Krattenthaler, University of ViennaBrandt Kronholm, University Of Texas Rio Grande ValleyRochelle Kronzek, World Scientific PublishersKag^an Kurşungö, Sabanci University, IstanbulJeff Lagarias, University of MichiganRobert Lemke Oliver, Stanford UniversityJim Lepowsky, Rutgers UniversityWinnie Li, Pennsylvania State UniversityZhi Guo Liu, East China Normal University, ShanghaiMaddie Locus, Emory UniversitySteffen Loebrich, University of CologneElizabeth Loew, SpringerLisa Lorentzen, Norwegian University of Science and TechnologyKarl Mahlburg, Louisiana State UniversityHelmut Maier, Universität UlmBibekananda Maji, Harish-Chandra Research Institute, AllahabadAmita Malik, University of IllinoisWenjun Ma, Shandong UniversityJames Maynard, University of OxfordKamel Mazhouda, Universite de Monastir, TunisiaRichard McIntosh, University of ReginaJim McLaughlin, West Chester University of PennsylvaniaStephen C Milne, Ohio State UniversityAdriana Morales, University of Puerto RicoTodd Molnar, University of FloridaHugh Montgomery, University of MichiganMichael Mossinghoff, Davidson CollegeAnton Mosunov, University of WaterlooK. A. Muttalib, University of FloridaMel Nathanson, Lehman College, CUNYKen Ono, Emory UnversityDonny Passary, The Pennsylvania State UniversityFrank Patane, Samford University

List of Participants xix

Peter Paule, RISC, J. Kepler University, LinzJohn Pfeilsticker, University of FloridaJanos Pintz, MTA Rényi Institute, Budapest, HungaryPaul Pollack, University of GeorgiaB. Ramakrishnan, Harish-Chandra InstituteSamin Riasat, University of WaterlooMichael Robinson, Center for Computing Sciences, MarylandSinai Robins, Brown University and ICERMLarry Rolen, Pennsylvania State UniversityMohan Rudravarapu, Government Polytechnic, Visakhapatnam, AP,IndiaCarla Savage, North Carolina State UniversityMichael Schlosser, University of ViennaRobert P. Schneider, Emory UniversityJames Sellers, Pennsylvania State UniversityLi Shen, University of FloridaAndrew Sills, Georgia Southern UniversityPeter Sin, University of FloridaMeera Sitharam, University of FloridaJaebum Sohn, Yonsei University, SeoulKannan Soundararajan, Stanford UniversityRichard Stanley, University of MiamiCameron Stewart, University of WaterlooArmin Straub, University of South AlabamaMarc Strauss, SpringerSergei Suslov, Arizona State UniversityKaren Taylor, CUNYGérald Tenenbaum, Université de LorraineJesse Thorner, Emory UniversitySarah Trebat-Leder, Emory UniversityLee Troupe, University of GeorgiaAlexandre Turull, University of FloridaAli Uncu, University of FloridaRobert Vaughan, Pennsylvania State UniversityChristophe Vignat, Tulane UniversityIan Wagner, Emory UniversityTanay Wakhare, Quince Orchard High School, Gaithersburg, MarylandLiuquan Wang, National University of SingaporeOle Warnaar, The University of Queensland, Australia

xx List of Participants

Victor Weenink, Radboud University NijmegenMichael Woodbury, University of CologneStanley Yao Xiao, University of WaterlooRuize Yang, University of FloridaAe Ja Yee, Pennsylvania State UniversityDoron Zeilberger, Rutgers UniversityWadim Zudilin, University of Newcastle, Australia

Conference Photographs

Krishna Alladi speaking at the Conference Banquet

List of Participants xxi

International Conference on Number Theory in honor of Krishna Alladi’s 60th Birthday,Gainesville, Florida, March 17–21, 2016

2003 SASTRA Ramanujan Prize Winner and 2014 Fields Medalist Manjul Bhargava (PrincetonUniversity) giving the Opening Lecture of the conference

xxii Conference Photographs

A section of the audience at the Straughn Center in between lectures. Peter Paule (L) and GeorgeAndrews (R) are in the front row and Ken Ono (second row left) is right behind them

Krishna Alladi (center) flanked by the conference organizers Frank Garvan (L) and GeorgeAndrews (R)

Conference Photographs xxiii

Krishna Alladi with Doron Zeilberger (Rutgers University) after Doron’s exciting lecture

After 2014 SASTRA Ramanujan Prize Winner James Maynard (University of Oxford) deliveredthe Ramanujan Colloquium, he accepts a cheque for $5,000 from Ron Graham (University ofCalifornia San Diego) for his work on the resolution of the problem posed by Paul Erdős on largegaps between primes

xxiv Conference Photographs

Christian Krattenthaler (University of Vienna) plays for the conference participants during theWelcome Reception at the Keene Faculty Center

Conference participants at the Welcome Reception at the elegant Keene Faculty Center

Conference Photographs xxv

Richard Askey (University of Wisconsin) speaking at the Welcome Reception

Kevin Ford (University of Illinois, Urbana) about to accept a cheque for $5,000 from Ron Graham(University of California San Diego) on behalf of his co-authors Ben Green, S. Konyagin, andTerence Tao for their work on the resolution of the $10,000 problem of Paul Erdős on large gapsbetween primes

xxvi Conference Photographs

Hugh Montgomery (University of Michigan), Krishna’s postdoctoral mentor, delivering the ErdősColloquium

Conference Photographs xxvii

(L to R) Michael Hirschhorn (University of New South Wales, Australia) and James Sellers(Pennsylvania State University) discussing during a break between sessions at the Straughn Center

(L to R) Doron Zeilberger (Rutgers University) discussing with University of Florida students

xxviii Conference Photographs

GauravBhatnagar (L) andOleWarnaar (R) enjoying a point beingmade byPeter Paule (RISC,Austria)

Manjul Bhargava (Princeton University) with the Alladi family at the Alladi home during theconference party. From L to R: daughter Lalitha Alladi, V. R. Srinivasan, wife Mathura Alladi,Manjul Bhargava, Krishna’s mother Lalitha Ramakrishnan, son-in-law Jis Joseph, and Krishna

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Conference participants on the deck at Alladi’s home during the Conference Party

(L to R) Ron Graham (University of California San Diego), Aleksandar Ivic (University ofBelgrade), Jean-Marc Deshouillers (University of Bordeaux), and Mel Nathanson (LehmanCollege, CUNY) in a discussion as they enjoy an Indian dinner at the Alladi House

xxx Conference Photographs

(L to R) Nick Diculeanu (Krishna’s colleague), Sinai Robins (Brown University), and MichaelSchlosser (University of Vienna) relaxing at the Conference Party at the Alladi House

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