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Celebratio Mathematica

Paul T. Bateman

Number theory  ·  UIUC

Memories of four decades with P. T. B.

by Harold G. Diamond

I came to Illinois in the Great Mi­gra­tion of the 1960s, as part of the ex­pan­sion and re­new­al of the Illinois Math De­part­ment. In my case the at­trac­tion was neither the scenery nor the cli­mate, but the pres­ence of Paul Bate­man. We shared in­terests in prime num­bers, tauberi­an and os­cil­la­tion the­or­ems, and sev­er­al oth­er top­ics. Paul had a sig­ni­fic­ant ef­fect on me and many oth­ers, through his in­terest, activ­ity and broad know­ledge.

Paul al­ways had pro­jects go­ing, and I be­came a part of many of them. These in­cluded or­gan­iz­ing con­fer­ences, jointly writ­ing half-a-dozen art­icles and a book on ana­lyt­ic num­ber the­ory [2] com­pleted when Paul was 85 years old, and tak­ing on a 5-year stint as co-ed­it­ors of the Prob­lem Sec­tion of the Amer­ic­an Math­em­at­ic­al Monthly. Paul was not one to palm off un­pleas­ant work on ju­ni­or part­ners — for the ed­it­or­ship, for ex­ample, he did all the large amount of work of writ­ing to au­thors and ref­er­ees, and I just stud­ied the suit­ab­il­ity of pro­posed prob­lems, which was much more fun. He was very proud of our hav­ing cleaned up a back­log of un­solved prob­lems and, as he put it, of “keep­ing egg off our faces.”

While pre­par­ing a sur­vey art­icle on Beurl­ing gen­er­al­ized num­bers [1], Paul and I de­veloped a con­jec­ture that the prime num­ber the­or­em for these num­bers held un­der an \( L^2 \) hy­po­thes­is weak­er than Beurl­ing’s fam­ous point­wise con­di­tion. This con­jec­ture was proved some 30 years later by Jean-Pierre Ka­hane in a series of im­press­ive art­icles, which were cited in his elec­tion as a full mem­ber of the French Academy of Sci­ences in 1998.

Along with his oth­er col­leagues and friends, I will re­mem­ber Paul for his guid­ance, sup­port, en­cour­age­ment, and friend­ship.

Works

[1]P. T. Bate­man and H. G. Dia­mond: “Asymp­tot­ic dis­tri­bu­tion of Beurl­ing’s gen­er­al­ized prime num­bers,” pp. 152–​210 in Stud­ies in num­ber the­ory. Edi­ted by W. J. LeVeque. MAA Stud­ies in Math­em­at­ics 6. Pren­tice-Hall (Engle­wood Cliffs, NJ), 1969. MR 0242778 Zbl 0216.​31403 incollection

[2]P. T. Bate­man and H. G. Dia­mond: Ana­lyt­ic num­ber the­ory: An in­tro­duct­ory course. Mono­graphs in Num­ber The­ory 1. World Sci­entif­ic (River Edge, NJ), 2004. MR 2111739 Zbl 1074.​11001 book