Let $f(n)=\sigma(n)/e^\gamma n\log\log n$, $n=3,4,\ldots$ , where $\sigma$ denotes the sum of divisors function. In 1984 Robin proved that the inequality $f(n)>1$, for all $n\ge 5041$, is equivalent to the Riemann hypothesis. Here we show that the values of $f$ are close to $0$ on a set of asymptotic density $1$. Similarly, an inequality by Rosser and Schoenfeld of 1962, dealing with the Euler totient function $\varphi$, is essential only on "thin" subsets of $\mathbf{N}$.

We obtain a Clunie type theorem for a rather general form of functional equations involving differential polynomials. Our theorems can give a much sharper estimate on the error term of the proximity function of solutions of differential equations and functional equations than the upper bound obtained by Clunie, Doeringer, He-Xiao, Korhonen and etc. In particular, our theorem can also be applied to study various types of Painlevé differential equations.

Let $G$ be a separable locally compact unimodular group of type I, $ \widehat{G}$ be its dual, $\hat{p}$ is a measurable field of, not necessary bounded, operators on $\widehat{G}$ such that $\hat{p}(\pi)$ is self-adjoint, $\hat{p}(\pi) \geq I$ for $\mu-$almost all $\pi \in \widehat{G}$, and \begin{align*} & A_{\hat{p} }(G) =\{f(x):=\int_{ \widehat{G}} Tr[\hat{f}(\pi)\pi(x)^{-1}]d\mu(\pi), \hat{f} \in L_{1}( \widehat{G} ), \|f\|_{\hat{p}} \\& \qquad =\int_{ \widehat{G} }Tr|\hat{p}(\pi)\hat{f}(\pi)|d\mu(\pi) >\infty \} \end{align*} We show that $ A_{\hat{p} }(G)$ is a Banach space endowed with the norm $\|f\|_{\hat{p}}$, and we generalize this result to the matricial group $G=G_{nm}$, $m\geq n$, of a local field.

Let $\mathcal F$ be a family of all functions $f$ meromorphic in a domain $D\subset\Bbb C$, for which, all zeros have multiplicity at least $k$, and $f(z)=0\Leftrightarrow f^{(k)}(z) = 1\Rightarrow |f^{(k+1)}(z)|\le h$, where $k\in\Bbb N$ and $h\in\Bbb R^+$ are given. Examples show that $\mathcal F$ is not normal in general (at least for $k=1$ or $k=2$). The example we give for $k = 1$ shows that a recent result of Y. Xu [5] is not correct. However, we prove that for $k\not=2$, there exists a positive integer $K\in\Bbb N$ such that the subfamily $\mathcal G =\{ f\in\mathcal F:\ \text{all possible poles of}\ f\ \text{in}\ D\ \text{have multiplicity at least}\ K\}$ of $\mathcal F$ is normal. This generalizes our result in [1]. The case $k = 2$ is also considered.