#### 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 __\( s \)__ 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

__\( n \)__is of the form__\( 2^k\pm1 \)__or__\( 4^k\pm 3 \)__, and__\( (2^n+1)/3 \)__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.