Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_k\left(x\right)$ where $1/k!{\sum }_{n\le x}n/\varphi \left(n\right){(1-n/x)}^k=M_k\left(x\right)+E_k\left(x\right)$. For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k\gg {\left(logx\right)}^{1+ϵ}$.

We give a method of obtaining explicit formulas for various mean values of Dirichlet L-functions which are expressed in terms of the Riemann zeta-function, the Euler function and Jordan's totient functions. Applying those results to mean values of Dirichlet L-functions, we also give an explicit formula for certain mean values of double Dirichlet L-functions.

Let $J_k\left(n\right):=n^k{\prod }_{p\mid n}(1-p^{-k})$ (the $k$-th Jordan totient function, and for $k=1$ the Euler phi function), and consider the associated error term$$E_k\left(x\right):=\sum _{n\le x}{J}_k\left(n\right)-\frac{x^{k+1}}{(k+1)\zeta (k+1)}.$$When $k\ge 2$, both $i_k:=E_k\left(x\right)x^{-k}$ and $s_k:=lim\; sup{E}_k\left(x\right)x^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead$$Ik:=\underset{n\in \mathbb{N},n\to \infty }{lim\; inf}$$d 1 (d)dk ( 12 - { nd} ), since from [AS] $i_k=I_k-{\left(\zeta (k+1)\right)}^{-}1$ and from the present paper $s_k=-i_k$. We show that $I_k$ belongs to an interval of the form$$\left(\frac{1}{2\zeta \left(k\right)}-\frac{1}{(k-1)N^{k-1}},\frac{1}{2\zeta \left(k\right)}\right),$$where $N=N\left(k\right)\to \infty$ as $k\to \infty$. From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of $I_k$. We apply this algorithm...