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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...

The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{4/3}$ for the error term $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{1+ϵ}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical...

I give explicit values for the constant implied by an Omega-estimate due to Chen and Chen [CC] on the average of the sum of the divisors of n which are relatively coprime to any given integer a.

The functional-differential equation $f^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ is closely related to Golomb’s self-described sequence $F$, $$\underset{1,}{\underbrace{1,}}\underset{2,}{\underbrace{2,2,}}\underset{2,}{\underbrace{3,3,}}\underset{3,}{\underbrace{4,4,4}}\underset{3,}{\underbrace{5,5,5,}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$$ We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number $p\ge 0$ there is exactly one increasing solution with $p$ as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely...