For a wide class of Young functions $\Phi:[0,\infty)\to[0,\infty)$, we determine the best constant $C_{\Phi}$ such that the following holds. If $(h_{k})_{k\geq 0}$ is the Haar system on $[0,1]$, then for any vectors $a_{k}$ from a separable Hilbert space $\mathcal{H}$ and $\varepsilon_{k}\in \{-1,1\}$, $k=0, 1, 2,\ldots$, we have \begin{equation*} \left|\left\{x\in [0,1]:\left|∑_{k=0}^{n} ɛ_{k}a_{k}h_{k}(x)\right|≥ 1\right\}\right|≤ C_{Φ} ∫_{0}^{1}Φ\left(\left|∑_{k=0}^{n} a_{k}h_{k}(x)\right|\right)\mathrm{d}x,\quad n=0,1,2,…. \end{equation*} This is generalized to the sharp weak-$\Phi$ inequality \begin{equation*} \mathbf{P}(\sup_{t≥ 0}|Y_{t}|≥ 1)≤ C_{Φ}\sup_{t≥ 0}\mathbf{E} Φ(|X_{t}|), \end{equation*} where $X$, $Y$ stand for $\mathcal{H}$-valued martingales such that $Y$ is differentially subordinate to $X$. These statements complement and generalize the results of Burkholder, Suh, the author and others.

In this article we announce fundamental results of Selberg type zeta functions for the Hilbert modular group of a real quadratic field; the meromorphic extension over $\mathbf{C}$, its functional equation and some arithmetic applications.

For a normal subgroup $K$ of a finite group $G$ and a $G$-invariant irreducible character $\xi$ of $K$ we show under a certain condition there is a bijection between the set of relative defect zero irreducible characters of $G$ lying over $\xi$ and the set of defect zero irreducible characters of $G/K$.

We study the weak convergence of some measures related to the renewal theorem, extending a result by Feller and Orey.

In this article, we give a new lower bound for the dimension of the linear space over the rationals spanned by 1 and values of polylogarithmic functions at a non-zero rational number. Our proof uses Padé approximation following the argument of T. Rivoal, however we adapt a new linear independence criterion due to S. Fischler and W. Zudilin. We also present an example of the linear space of dimension $\geqslant 3$ over $\mathbf{Q}$, which is generated by 1 and polylogarithms.