Abstract

I will introduce some of my works in probabilistic number theory. Chebyshev (1853) noticed that there seems to be more primes congruent to 3 mod 4 than that congruent 1 mod 4. This is known as “Chebyshev’s bias” problem, or “prime number races” problem. Under certain reasonable conditions, Rubinstein and Sarnak (1994) gave quantitative result for this phenomenon. I generalized the Chebyshev’s bias problem to the distribution of products of k primes among different arithmetic progressions. In the case of counting products of primes with multiplicity, the direction of the Chebyshev’s bias depends on the parity of k. And if we count primes without multiplicity, certain residue classes always dominate. A related problem is to consider products of primes of which each prime factor is from certain arithmetic progressions. In this case, the “bias” is much larger than the original “Chebyshev’s bias” problem. If time permits, I will introduce my recent work on visible lattice points in random walks.