Celebratio Mathematica

Paul was broadly edu­cated in num­ber the­ory. His re­search centered on clas­sic­al ana­lyt­ic num­ber the­ory and as­so­ci­ated ana­lys­is. Top­ics in­cluded sums of squares, mod­u­lar forms, the dis­tri­bu­tion of prime num­bers, Beurl­ing’s gen­er­al­ized prime num­bers, geo­met­ric­al ex­trema, coef­fi­cients of cyc­lo­tom­ic poly­no­mi­als, and arith­met­ic­al func­tions. Paul wrote joint pa­pers with over 20 dif­fer­ent coau­thors, and su­per­vised 20 doc­tor­al stu­dents in num­ber the­ory, in­clud­ing the late Mar­vin Knopp of Temple Uni­versity and Kev­in Mc­Cur­ley of Google.

Paul’s first ma­jor res­ult was con­tained in his thes­is and pub­lished in the AMS Trans­ac­tions [1]. It proved a for­mula con­jec­tured by G. H. Hardy for the num­ber of rep­res­ent­a­tions of a pos­it­ive in­teger as the sum of three squares. Hardy had ob­tained ex­act for­mu­las for sums of \( s \) squares for \( 5 \le s \le 8 \). Paul was able to handle the case \( s=4 \) quite eas­ily, but \( s=3 \) re­quired a dif­fer­ent ar­gu­ment, sug­ges­ted by work of Erich Hecke. The key step in his ar­gu­ment was to es­tab­lish con­ver­gence of Hardy’s so-called sin­gu­lar series \( \mathfrak{S}_3(n) \) by a subtle lim­it­ing ar­gu­ment. Dozens of au­thors be­fore Paul had stud­ied the num­ber of rep­res­ent­a­tions of in­tegers as sums of squares, and in­terest in this top­ic and his work con­tin­ues — since the year 2000, Paul’s pa­per has been cited in 14 art­icles.

Paul is per­haps best known to the num­ber the­ory com­munity for the Bate­man–Horn con­jec­tur­al asymp­tot­ic for­mula for the num­ber of \( k \)-tuples of primes gen­er­ated by sys­tems of poly­no­mi­als [3]. (The most cel­eb­rated prob­lem of this type is the twin-prime con­jec­ture; an­oth­er is to de­term­ine how of­ten \( n^2 + 1 \) is a prime.) Their for­mula ex­ten­ded and quan­ti­fied sev­er­al fam­ous con­jec­tures of Hardy and J. E. Lit­tle­wood, and of An­drzej Schin­zel, and they il­lus­trated its qual­ity with cal­cu­la­tions. This top­ic has been treated in dozens of sub­sequent pa­pers.

An­oth­er of Paul’s com­pu­ta­tion-re­lated pro­jects dealt with Mersenne primes. These are prime num­bers of the form \( M_n=2^n-1 \). The ques­tion of the prim­al­ity of num­bers \( M_n \) has fas­cin­ated num­ber the­or­ists for cen­tur­ies. (The UIUC Math De­part­ment postal stamp in the 1970s an­nounced that the Mersenne num­ber \( 2^{11,213} -1 \) is prime.) A ne­ces­sary con­di­tion for \( M_n \) to be prime is that \( n \) be prime, but bey­ond this ele­ment­ary res­ult little is known. There are fam­ous con­jec­tures of wheth­er in­fin­itely many Mersenne primes ex­ist, and there have been large-scale com­pu­ta­tion­al ef­forts to find large Mersenne primes.

In 1989, Bate­man, John Sel­fridge and Samuel Wag­staff [5] for­mu­lated a “New Mersenne Con­jec­ture,” to cor­rect Mar­in Mersenne’s ori­gin­al (flawed) prim­al­ity claims. The new con­jec­ture states that if

1. \( n \) is of the form \( 2^k\pm1 \) or \( 4^k\pm 3 \), and

2. \( (2^n+1)/3 \) is prime,

then \( M_n \) is prime; con­versely, if \( M_n \) is prime and one of the con­di­tions (1) and (2) holds, then the oth­er holds as well. This con­jec­ture has been veri­fied for \( p\le 2\cdot 10^7 \) [e2]. Al­though it may be re­garded as mainly a curi­os­ity, the con­jec­ture has con­tin­ued to gen­er­ate in­terest over the years. It is dis­cussed in art­icles on Wiki­pe­dia,1 in Eric Weis­stein’s Math­World web­site [e1], as well as in Chris Cald­well’s Prime Pages site [e2].

Paul had an en­cyc­lo­ped­ic know­ledge of the lit­er­at­ure of num­ber the­ory. In ad­di­tion to serving as a mo­bile ver­sion of Math Re­views, he put this tal­ent to good use in writ­ing an au­thor­it­at­ive ap­pendix for the re­print of Ed­mund Land­au’s ground­break­ing 1909 book Primzah­len [2].

An­oth­er of Paul’s en­thu­si­asms was prob­lem solv­ing. He in­spired gen­er­a­tions of stu­dents and en­riched sev­er­al books with his prob­lems. Throughout his ca­reer, Paul con­trib­uted nu­mer­ous prob­lems and solu­tions to the Amer­ic­an Math­em­at­ic­al Monthly: 56 of his con­tri­bu­tions ap­pear in print. It was not sur­pris­ing that, among his sev­er­al ed­it­or­ships, he served as a coed­it­or of the Prob­lems Sec­tion.

One of his prob­lems, cre­ated jointly with Bruce Reznick (sub­mit­ted un­der the pseud­onym P. A. Bat­nik) [4] re­flec­ted their com­mon in­terest in sums of squares: Prove that if \( n \) is an odd pos­it­ive in­teger, there ex­ist in­tegers \( x_1,\ x_2,\ x_3,\ x_4 \) with \[ n = x_1^2 + x_2^2 + x_3^2 + x_4^2 \qquad\text{and}\qquad x_1 \ge \lfloor \sqrt{n} \rfloor - 1 .\]

From 1965 to 1980, Paul served as de­part­ment head at UIUC. This was a time of ma­jor ex­pan­sion and fac­ulty re­new­al, and Paul raised the level of the whole de­part­ment. His en­er­get­ic lead­er­ship and his pro­mo­tion of sem­inars and con­fer­ences helped form an out­stand­ing num­ber the­ory group, and had a pos­it­ive ef­fect on the sub­ject throughout the na­tion. Paul had both good ideas and the drive to carry them out.

Paul’s ca­pa­city for work was im­press­ive. As head, he was single-handedly re­spons­ible for nearly all the de­part­ment’s ad­min­is­trat­ive work, in­clud­ing pre­par­ing doc­u­ments on hir­ing and pro­mo­tion, NSF grant ad­min­is­tra­tion, pre­par­ing budgets, and re­con­cil­ing ex­pendit­ures. Dur­ing this time, he was also an As­so­ci­ate Sec­ret­ary of the AMS. In ad­di­tion, Paul main­tained an act­ive re­search pro­gram, su­per­vised gradu­ate stu­dents, and made time to sup­port tal­en­ted young people.