Integers free of small prime factors in arithmetic progressions

Source abstract

For real x ≥ y ≥ 2 and positive integers a, q , let Φ( x, y; a, q ) denote the number of positive integers ≤ x , free of prime factors ≤ y and satisfying n ≡ a (mod q ). By the fundamental lemma of sieve, it follows that for ( a,q ) = 1, Φ( x,y;a,q ) = φ( q )- 1 , Φ( x, y ){1 + O (exp(-u(log u - log 2 3 u - 2))) + ( u = log x log y ) holds uniformly in a wider ranges of x, y and q . Let χ be any character to the modulus q , and L(s, χ ) be the corresponding L -function. Let be a (‘exceptional’) real character to the modulus q for which L(s , ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q . In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u ) is Dickman function, and ρ′(u ) = dρ(u)/du . The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q) A .