Duality between prime factors and an application to the prime number theorem for arithmetic progressions

An analogue of a formula for Chebotarev densities

“…For example, one can get a version of Theorem 3.1 for P m (n). Another example we would like to mention is that P m (n) is equi-distributed (mod k) for k 2 by Theorem 1 in [2]. Now we prove the Theorem 1.1 by showing the following theorem.…”

“…Let p(n) be the smallest prime divisor of n. Let k 1, ℓ be integers and (ℓ, k) = 1. In 1977, Alladi [2] proved that − n 2 p(n)≡ℓ(mod k) µ(n) n = 1 ϕ(k) .…”

“…So λ(n) = d|n µ(d)a( n d ) and a(n) = d|n λ m (d) by Möbius inversion formula. Following [2], for n = p α1 1 · · · p αr r , p 1 < · · · < p r , we have…”

Möbius formulas for densities of sets of prime ideals

“…In particular, let p min (n) denote the smallest prime divisor of an integer n ≥ 2. Alladi [1] proved − n≥2 p min (n)≡a (mod q) µ(n) n = 1 ϕ(q) for any positive integer q and integer a with gcd(a, q) = 1.…”

Asymptotic Estimates of Sums Involving the Moebius Function. II

“…and has an interesting application to the Prime Number Theorem for Arithmetic Progressions. For details see [1].…”