The Distribution of Certain Restricted Numbers
Xianchang Meng, Centre de Recherches Mathématiques in Montréal, Canada
【Abstract】Chebyshev noticed that there seems to be more number of primes congruent to 3 mod 4 than those congruent to 1 mod 4. Questions related to the distribution of prime numbers among different arithmetic progressions are known as ``Prime Race Problems". I will introduce some generalizations of the prime number races: 1) the distribution of products of k primes in different arithmetic progressions; the results are different if we count the number of prime factors with multiplicity or not; 2) a generalization of a very recent result of Dummit, Granville, and Kisilevsky who studied the distribution of products of two primes pq with p, q both from the residue class 3 mod 4; 3) Function field version of prime number races. Probabilistic method is a very useful tool to study prime number races. If time permits, I may mention how to improve the error term in the counting function of k-free numbers using probabilistic method under some reasonable conjectures.