Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while $\theta \left(x\right)={\sum }_{p\le x}logp$. The best known inequalities involving the function π(x) are the ones obtained in [6] by B. Rosser and L. Schoenfeld: (1) x/(log x - 1/2) < π(x) for x ≥ 67 (2) x/(log x - 3/2) > π(x) for $x>e^{3/2}$. The proof of the above inequalities is not elementary and is based on the first 25 000 zeros of the Riemann function ξ(s) obtained by D. H. Lehmer [4]. Then Rosser, Yohe and Schoenfeld...

Let $$T\left(q\right)=\sum _{k=1}^{\infty }d\left(k\right)q^k,\left|q\right|<1,$$ where $d\left(k\right)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$H\left(q\right)=T\left(q\right)-\frac{log(1-q)}{log\left(q\right)}$$ is strictly increasing on $(0,1)$, while $$F\left(q\right)=\frac{1-q}{q}H\left(q\right)$$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)}<T\left(q\right)<\beta \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)},0<q<1,$$ holds with the best possible constant factors $\alpha =\gamma$ and $\beta =1$. Here, $\gamma$ denotes Euler’s constant. This refines a result of Salem, who proved the inequalities...

For positive integers $n$, Euler’s phi function and Dedekind’s psi function are given by $$\phi \left(n\right)=n\prod _{\begin{array}{c}p\mid n\\ p\mathrm{prime}\end{array}}\left(1-\frac{1}{p}\right)\text{and}\psi \left(n\right)=n\prod _{\begin{array}{c}p\mid n\\ p\mathrm{prime}\end{array}}\left(1+\frac{1}{p}\right),$$ respectively. We prove that for all $n\ge 2$ we have $${\left(1-\frac{1}{n}\right)}^{n-1}{\left(1+\frac{1}{n}\right)}^{n+1}\le {\left(\frac{\phi \left(n\right)}{n}\right)}^{\phi \left(n\right)}{\left(\frac{\psi \left(n\right)}{n}\right)}^{\psi \left(n\right)}$$ and $${\left(\frac{\phi \left(n\right)}{n}\right)}^{\psi \left(n\right)}{\left(\frac{\psi \left(n\right)}{n}\right)}^{\phi \left(n\right)}\le {\left(1-\frac{1}{n}\right)}^{n+1}{\left(1+\frac{1}{n}\right)}^{n-1}.$$ The sign of equality holds if and only if $n$ is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).

For any positive integer $n$ let ϕ(n) be the Euler function of n. A positive integer $n$ is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ${\left(2^{m}k\right)}_{m\ge 1}$ consists entirely of noncototients. We then use computations to detect seven such positive integers k.