# 1993 Comparative Number Theory

Topics in comparative number theory (Giuliana Davidoff).

Davidoff's students were Caroline Osowski (Mount Holyoke '93, M.A. in Mathematics, Mount Holyoke 1994), Yi Wang (Bryn Mawr '94, who became a graduate student in the Ph.D. program in Operations Research at MIT), Jennifer vanden Eynden (University of Illinois '94, who became a graduate student in the Ph.D. program in Mathematics at the University of California, San Diego), and Nancy Wrinkle (Barnard '94).

The project examined some old and some new questions about distributions of primes among residue classes with a given difference k. Define a function \pi(x,k,a) which counts prime numbers in the arithmetic progression a + kd by \pi(x,k,a) = # { p less than or equal to x such that p is a prime and p = a+kd for some integer d }. In the first and most widely recognized result in the study of finer comparisons of primes among progressions, Littlewood proved that the difference \pi(x,4,3) - \pi(x,4,1) changes sign infinitely often as x becomes large. About 50 years later Knapowski and Turan asked, among other things, whether this same oscillatory behavior exists for any residue classes a and b for a given arbitrary modulus k. They were able to answer many questions, but the general result remained unknown, even for small k.

Our research project focused on further investigations both of the generalization introduced by Knapowski and Turan and of our own apparently new one which asked about the oscillatory behavior of another difference function. Using further work of Bays, Hudson, Spira, and Stark on the first generalization, along with new calculations of zeroes of L-functions done recently by Rumely together with our own numerical work, we were able to prove that this new difference changes sign infinitely often in many cases. Extending the work of this summer research project, Peter Sarnak and his thesis student, Michael Rubenstein, have recently settled Knapowski and Turan's question in proving, among other results, that these difference functions change sign infinitely often.