Sums Over Primes II

Abstract

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.

Sums over primes III

“…The motivation of this paper arises from an exercise in Granville's book [5] and the author's recent work [14] on generalized alternating hyperharmonic numbers H (p,r,2,1) n and H (p,r,1,2) n . This paper is a continuation of the previous papers of the author with the same title [12,15]. In this paper, we will derive explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types H (p,r,2,1) n and H (p,r,1,2) n .…”

“…Lemma 3 ( [7,12]). Let p n,k denote the nth squarefree number with just k prime factors and q n,k denote the nth number with k prime factors counted with multiplicity.…”

“…The author [12,15] considered the sum involving generalized hyperharmonic numbers of type pn≤x p α n (H (p,r) n…”