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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.
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 $X -> oo$,
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