A Generalization Of Bombieri-Vinogradov Type Theorem

About 200 yeas ago, Legendre and Gauss found independently that the prime numbers up to x is - x/logx for large x, this is known as the prime number theorem. This assertion was proved independently in 1896 by Hadamard and de la Vallee Poussin. An equivalent statement of the prime number theorem is:ψ(x) - x, whereψ(x) = (?). Actually, an asymptotic formula holds, i.e.Under Riemann's hypothesis (RH in brief), the above error term becomes x^1/2 log² x.As for prime number theorem in arithmetic progressions, it was stated that for relatively prime integers a and q,whereThis assertion is not proved for 'large' q. But for 'small' q, Siegel-Walfisz theorem states that for arbitrary A > 0, q = O(log^A x), However, under GRH, the above error term is O(x^1/2log² x), and hence (1) holdsfor q=O(x^1/2log^-(2+ε) x).Bombieri-Vinogradov theorem states that for any fixed A > 0, there exist a constant B = B(A) such thatfor Q = x^1/2(log x)^-B.The Bombieri-Vinogradov theorem is an amazingly strong unconditional replacement for the (1) on average. It says that if you pick out the worst error term modulo q for each q up to about x^1/2, and add these up, you get roughly what GRH predicts.In this paper, we consider Bombieri-Vinogradov type theorem for k-almost primes for positive integer k. LetΛ_k be the generalized von Mangoldt function defined byΛ_k =μ*log^k, i.e.Definewhich is known as Selberg function. Obviouslyψ₁(x;q,a) =ψ(x;q,a). For k≥2, one seeks result like (3) forψ_k(x;q,a), preciselyIn fact,Thus, For k = 2, by using the sieve method developed by Bombieri ([2],[3]) and ([5]), Friedlander ([4]) proved the following:Theorem 1 (Friedlander) Let a and q be relatively prime positive integers. Letε(x) be a fixed positive function, tending to 0 as x→∞. The asymptotic formulaholds uniformly in the rangeFor k≥2, Emmanuel Knafo ([1]). generalized Fridlander's result and proved thatTheorem 2 (Emmanuel Knafo) Let k≥2. Letε(x) be a fixed positive function, tending to 0 as x→∞. The asymptotic formulaholds uniformly in the rangeIn Theorems 1 and 2, we see that (7) holds only for 'small' q.In this paper, we will give a theorem similar to Bombieri-Vinogradov theorem forψ_k(x; q, a). Our result predicts that (7) holds on average for q≤Q = x^1/2log^-B x.For some technical reason, we can't proveThe main theorem in this paper is the following.Theorem Let k, a, q be positive integers with k≥1, (a,q) = 1. Let x > 0 be a large real number. Then for any fixed A>0, there exist a constant B=B(A) such that holds for Q = (?).