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 $2$.

As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh\left(x\right)$ where $h$ is an arithmetical function (namely $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) and $n$ is an integer (or a prime order) running over the interval $\left[y\right(x),x)]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

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 ).

To a plurisubharmonic function $u$ on ${\mathbf{C}}^n$ with logarithmic growth at infinity, we may associate the Robin function $${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$ defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ and (respectively) for which the function ${\rho }_u$ is not identically $-\infty$. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. As an application we obtain a criterion...

The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{4/3}$ for the error term $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{1+ϵ}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical -...

Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as $$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$ where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then $${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$ moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows ...

Canonical number systems in the ring of gaussian integers $\mathbb{Z}\left[i\right]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb{Z}\left[i\right]$. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$. In this paper we investigate the sum of digits function ${\nu }_b$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$-th power of a prime. Furthermore, we establish an Erdös-Kac type...