by Harold G. Diamond
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
Paul is perhaps best known to the number theory community for the
Bateman–Horn
conjectural asymptotic formula for the number of
Another of Paul’s computation-related projects dealt with Mersenne
primes. These are prime numbers of the form
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
is of the form or , and is prime,
then
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
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.