Let $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2$, be a Cantor scale, ${𝐙}_Q$ the compact projective limit group of the groups $𝐙/Q_k𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/q_j𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1$, of ${𝐙}_Q$ we associate the sequence of integral valued random variables $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. The main result of this article is that, given a complex $𝐐$-multiplicative function $g$ of modulus $1$, we have$$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

A new class of $b$-adic normal numbers is built recursively by using Eulerian paths in a sequence of de Bruijn digraphs. In this recursion, a path is constructed as an extension of the previous one, in such way that the $b$-adic block determined by the path contains the maximal number of different $b$-adic subblocks of consecutive lengths in the most compact arrangement. Any source of redundancy is avoided at every step. Our recursive construction is an alternative to the several well-known concatenative...

We show that the Deligne formal model of the Drinfeld $p$-adic half-plane relative to a local field $F$ represents a moduli problem of polarized $O_F$-modules with an action of the ring of integers in a quadratic extension $E$ of $F$. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of ${\mathrm{SL}}_2\left(F\right)$ and $\mathrm{SU}\left(C\right)\left(F\right)$ for a two-dimensional split hermitian space $C$ for $E/F$.

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for $\zeta \left(s\right)$. We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper “Remarques sur un beau rapport entre...

The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$