In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda =b_{2}log{\alpha }_2-b_{1}log{\alpha }_1-b_{3}log{\alpha }_3\ne 0$ with $b_1,b_2,b_3$ positive integers and ${\alpha }_1,{\alpha }_2,{\alpha }_3$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $(j=1,2,3)$. Let ${\alpha }_j\ge \text{max}\{{\alpha }_{j1},e\}$ and assume that gcd$(b_1,b_2,b_3)=1.$ Let $$b^{\text{'}}=\left(\frac{b_2}{log{\alpha }_1}+\frac{b_1}{log{\alpha }_2}\right)\left(\frac{b_2}{log{\alpha }_3}+\frac{b_3}{log{\alpha }_2}\right)$$ and assume that $B\ge \text{max}\{10,log{b}^{\text{'}}\}.$ We prove that either $\{b_1,b_2,b_3\}$ is $\left(c_4,B\right)$-linearly dependent over $\mathbb{Z}$ (with respect to $\left(a_1,a_2,a_3\right)$)...

The estimations of lower order $\left(R\right)$ $\lambda$ in terms of the sequences $\left\{a_n\right\}$ and $\left\{{\lambda }_n\right\}$ for an entire Dirichlet series $f\left(s\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{s\lambda n}$, have been obtained, namely : $$\begin{array}{cc}\hfill \lambda & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{{\lambda }_{n_p}log{\lambda }_{n_{p-1}}}{log|a_{n_p}{|}^{-1}}\hfill \\ & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{({\lambda }_{n_p}-{\lambda }_{n_{p-1}})log{\lambda }_{n_{p-1}}}{log\left|a_{n_{p-1}}\right|a_{n_p}|}.\hfill \end{array}$$ One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

We prove: If $A\left(n\right)$ and $G\left(n\right)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n\mapsto nA\left(n\right)/G\left(n\right)-(n-1)A(n-1)/G(n-1)$ $(n\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty$.

Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques...

For $x\in ℂ$, $\left|x\right|\<1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\<1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

Under the Generalized Riemann Hypothesis, it is proved that for any $k\ge 200$ there is $N_k\>0$ depending on $k$ only such that every even integer $\ge N_k$ is a sum of two odd primes and $k$ powers of $2$.