Abstract
This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.
Highlights
This paper is concerned with estimating the number of positive integers up to some bound, such that they have a fixed number of prime divisors, and lie in a given arithmetic progression
The special case of i) when n I is done in Rieger [4]
The lemma is trivial if k 1 Otherwise, let q denote the (k) th prime
Summary
As x tends to infinity, the sum on the left of (3.24) tends to zero, Izl uniformly in x, z, and X with o o I and. B. So, the sum in (3.14) converges uniformly in s, z, and X, as asserted. The quantity in (3.25) is uniformly convergent, as desired, and the lemma is proved in this case. The uniform convergence of the quantity in (3.14) can be derived from (3.17). In this case, the uniform convergence of the quantity in (3.25), and of the quantity in (3.15), follows from the fact that (pm) i whenever m 1 These derivations are similar to the argument that we have just made, and will be left to the reader. It follows from (2.7), (2.8), and equation (i) of Rieger [3] that n az (n’{)= 7. dz ()bz (n’. )’
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