Celebratio Mathematica

Paul T. Bateman

Number theory  ·  UIUC


by Hugh L. Montgomery

I went to the Uni­versity of Illinois for my un­der­gradu­ate stud­ies primar­ily be­cause it was re­puted to have a strong math­em­at­ics de­part­ment. I was already in­ter­ested in num­ber the­ory, and by blind luck I found my­self in a place with a strong pro­gram, which, un­der the lead­er­ship of Paul Bate­man, in­cluded my fa­vor­ite fla­vor of num­ber the­ory.

After ex­cel­lent in­tro­duct­ory hon­ors math courses, in the second semester of my sopho­more year I found my­self tak­ing Bate­man’s prob­lem sem­in­ar course. I was in way over my head! I spent 20 hours a week on that course alone while tak­ing a full aca­dem­ic load, but it was worth every minute. I learned about in­clu­sion-ex­clu­sion, sum­ma­tion by parts, abeli­an and tauberi­an the­or­ems, gen­er­at­ing func­tions, vari­ous kinds of con­ver­gence and the in­ter­change of double lim­its, lin­ear re­cur­rences, the Wei­er­strass ap­prox­im­a­tion the­or­em, the Pois­son sum­ma­tion for­mula, and uni­form dis­tri­bu­tion — to name just a few of the top­ics covered. I still refer to my prob­lem sem­in­ar coursep­ack and find it very valu­able.

Sub­sequently I took fur­ther courses from Bate­man, one on Di­o­phant­ine ap­prox­im­a­tion and the geo­metry of num­bers, and an­oth­er on ana­lyt­ic num­ber the­ory. The lat­ter course fea­tured a proof of the Prime Num­ber The­or­em for Beurl­ing primes.

Bate­man ar­ranged for vis­its from sev­er­al out­stand­ing math­em­aticians dur­ing my time as an un­der­gradu­ate. Thus I got the op­por­tun­ity to take a course from L. J. Mor­dell on di­o­phant­ine equa­tions, and a prob­lem-solv­ing course from Paul Er­dős and John Sel­fridge.

Paul had the idea of nom­in­at­ing me for a Mar­shall Schol­ar­ship, and he se­cured a let­ter from Har­old Dav­en­port of­fer­ing to take me as a stu­dent if I reached Cam­bridge. My good luck (and Paul’s strong sup­port) stayed with me, as I won the schol­ar­ship and got to work with Dav­en­port. Un­for­tu­nately, Dav­en­port had died by the time I fin­ished my thes­is; but around then Paul sug­ges­ted that I send the thes­is to Spring­er for their Lec­ture Notes series. It was ac­cep­ted and pub­lished as SLN 227 [e1].

Paul nom­in­ated me, a newly min­ted Ph.D., to serve as a mem­ber of the or­gan­iz­ing com­mit­tee of the large 1972 sym­posi­um on ana­lyt­ic num­ber the­ory, held at St. Louis Uni­versity. This was one of sev­er­al early op­por­tun­it­ies for me which be­came avail­able through Paul’s do­ing.

Paul con­tin­ued to be help­ful and sup­port­ive in many dif­fer­ent ways over the sub­sequent years. A con­crete in­stance of this took place just a few years ago, when I served on the Put­nam prob­lems com­mit­tee. When we were strug­gling to de­vise good prob­lems at the A-1 or B-1 level, Paul com­mu­nic­ated sev­er­al sol­id can­did­ates, drawn from his per­son­al col­lec­tion.

I am in­debted to Paul for the con­tinu­ing in­terest he took in my ca­reer, and the ini­ti­at­ive he ex­er­cised on my be­half.