题目：           Chebyshev's bias for products of k primes 

报告人：      Xianchang Meng  (Centre de Recherches Mathematiques in Montreal) 

时间地点：   2017.12.15  15:00pm  N202

摘要：           For any k _ 1, we derive a formula for the di_erence between the number of integers n _ x with !(n) = k or (n) = k in two di_erent arithmetic progressions, where !(n) is the number of distinct prime factors of n and (n) is the number of prime factors of n counted with multiplicity. Under some reasonable assumptions, we show that, if k is odd, the integers with (n) = k have preference for quadratic non-residue classes; and if k is even, such integers have preference for quadrat-ic residue classes. This result con_rms a conjecture of Hudson. However, the integers with !(n) = k always have preference for quadratic residue classes. Moreover, as k increases, the biases become smaller and smaller for both cases.