Let $\varphi$ denote the Euler function. In this paper, we show that for all large $x$ there are more than $x^{0.33}$ Carmichael numbers $n\le x$ with the property that $\varphi(n)$ is a perfect square. We also obtain similar results for higher powers.

The integers coprime to $n$ are called the {\it totatives} \rm of $n$. D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of the total number of totatives up to $n$. They provided criteria in various cases. Here we give an ``if and only if'' criterion which allows us to recover most of the previous results in this literature and to go beyond, as well to reformulate the problem in terms of combinatorial group theory. Our criterion is that the above holds if and only if for every odd character $\chi \pmod \kappa$ (where $\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$ dividing $n$ for which $\chi(p)=1$.

D.~H.~Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P.~Erd\H os, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.

This paper is a study of summability methods that are based on Dirichlet convolution. If $f(n)$ is a function on positive integers and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n} {1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$-summable\/} to $L$. The necessary and sufficient condition for the matrix $A_f$ to preserve bounded variation of sequences is established. Also, the matrix $A_f$ is investigated as $\ell - \ell$ and $G-G$ mappings. The strength of the $A_f$-matrix is also discussed.