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## Lester R. Ford Awards

The Lester
R. Ford Awards, established in 1964, are made to authors of
expository articles published in *The American Mathematical
Monthly. *The Awards are named for Lester R. Ford, Sr.,
a distinguished mathematician, editor of *The American
Mathematical
Monthly, *1942-46, and President of the Mathematical
Association
of America, 1947-48.

**Andrew
Granville and Greg Martin**

"Prime
Number Races," *American Mathematical Monthly,*

vol. 113, no. 1,2006, pp. 1-33.

The races
in this exciting article are between primes in different congruence
classes mod $q$. That is, fix $q$ and
consider for varying
values of integers $a$ relatively prime to $q$ the
functions $pi_(q,a) (x)=$ the number of primes of the
form $qn +
a$ less than or equal to $x$. The prime
number theorem for
arithmetic progressions tells us that, asymptotically, the results
for different such $a$ will be the same, but this
does not
address whether or how often $pi_(q,a) (x) >
pi_(q,b) (x)$ for
specific values of $x$.

Granville
and Martin offer a wonderful array of refined results about $pi_(q,a) (x)$ vs. $pi_(q,b) (x)$ beginning with
Chebychev's observation
that $pi_(4,3) (x)$ appears to be larger than $pi_(4,1) (x)$ coupled
with J. E. Littlewood's 1914 theorem that there are arbitrarily large
values of $x$ for which $pi_(4,1) (x) >
pi_(4,3) (x)$.
In addition, the authors highlight M. Rubinstein and P.
Sarnak's
amazing 1994 result that, as $X -> oo$,

$1/(ln X) sum_(x<=X, pi_(4,3)(x)>pi_(4,1)(x) ) 1/x -> .9959...$
Of course,
the generalized Riemann hypothesis figures in the discussion, as does
a survey of some very recent research including that of both authors,
as well as of teams of graduate and undergraduate (!) students.

Granville
and Martin ably capture the thrill of the chase, the mathematics, and
the many questions still to explore.

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*Last updated 26, August, 2007*