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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 πq,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 πq,a(x)>πq,b(x) for specific values of x.

Granville and Martin offer a wonderful array of refined results about πq,a(x) vs. πq,b(x) beginning with Chebychev's observation that π4,3(x) appears to be larger than π4,1(x) coupled with J. E. Littlewood's 1914 theorem that there are arbitrarily large values of x for which π4,1(x)>π4,3(x). In addition, the authors highlight M. Rubinstein and P. Sarnak's amazing 1994 result that, as X,

1lnXxX,π4,3(x)>π4,1(x)1x.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