Si associano ad una matrice infinita di un certo tipo altre due matrici dello stesso tipo, dette rispettivamente bernoulliana e antibernoulliana di A. Si studiano alcune proprietà di queste matrici. Si ottiene in tal via una generalizzazione dei classici numeri di Bernoulli.

Let $\xi =[a_0;a_1,a_2,\cdots ,a_i,\cdots ]$ be an irrational number in simple continued fraction expansion, $p_i/q_i=[a_0;a_1,a_2,\cdots ,a_i]$, $M_i=q_i^2|\xi -p_i/q_i|$. In this note we find a function $G(R,r)$ such that $$\begin{array}{ccc}& M_{n+1}<R\text{and}{M}_{n-1}<r\text{imply}{M}_n>G(R,r),\hfill & \hfill M_{n+1}>R\text{and}{M}_{n-1}>r\text{imply}{M}_n<G(R,r).\end{array}$$ Together with a result the author obtained, this shows that to find two best approximation functions $\tilde{H}(R,r)$ and $\tilde{L}(R,r)$ is a well-posed problem. This problem has not been solved yet.

It is proved that for almost all prime numbers $k\le N^{1/4-ϵ}$, any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying $p_i\equiv b_i\left(modk\right)$, i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.

This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f\left(s\right)={\sum }_{n=1}^{\infty }aₙn^{-s}$ with ${\left|\right|f\left|\right|}_{\infty }:=su{p}_{\Re s>0}\left|f\left(s\right)\right|<\infty$, then ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-2}{\le \left|\right|f\left|\right|}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-1/2}{\le C\left|\right|f\left|\right|}_{\infty }$, C being an absolute constant.