On the Riesz means of (Formula presented)$${\frac{n}{\phi(n)}}$$ n ϕ (n)-II(/Formula presented)

Abstract

Let (Formula presented)$${\phi(n)}$$ ϕ (n)(/Formula presented) denote the Euler-totient function. We study the error term of the general k-th Riesz mean of the arithmetical function $${\frac {n}{\phi(n)}}$$ n ϕ (n)(/Formula presented) for any positive integer $${k \ge 1}$$ k≥ 1(/Formula presented), namely the error term $${E_k(x)}$$ E k(x)(/Formula presented) where (Formula presented)$${\frac{1}{k!} \sum_{n \leq x} \frac{n}{\phi(n)} \left(1-\frac{n}{x}\right)^k = M_k(x) + E_k(x).}$$(/Formula presented)(Formula presented)1 k ∑ n ≤ x n ϕ (n) 1 n x k M k x + (x).(/Formula presented)The upper bound for (Formula presented)$${| E_k(x)|}$$ | E k(x) |(/Formula presented) established here thus improves the earlier known upper bounds for all integers (Formula presented)$${k\geq 1}$$ k ≥ 1(/Formula presented).