Dans cet article, nous donnons une minoration de la mesure de Mahler d'un polynôme à coefficients entiers, dont toutes les racines sont d'une part réelles positives, d'autre part réelles, en fonction de la valeur de ce polynôme en zéro. Ces minorations améliorent des résultats antérieurs de A. Schinzel. Par ailleurs, nous en déduisons des inégalités de M.-J. Bertin, liant la mesure d'un nombre algébrique à sa norme.

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).

In this note, particular inequalities of DVT-type in real and integer numbers are investigated.

Quadratic forms on a free finite-dimensional module are investigated. It is shown that the inertial law can be suitably generalized provided the vector space is replaced by a free finite-dimensional module over a certain linear algebra over $ℝ$ ( real plural algebra) introduced in [1].