Our previous research was devoted to the problem of determining the primitive periods of the sequences ${(G_{n}mod{p}^t)}_{n=1}^{\infty }$ where ${\left(G_n\right)}_{n=1}^{\infty }$ is a Tribonacci sequence defined by an arbitrary triple of integers. The solution to this problem was found for the case of powers of an arbitrary prime $p\ne 2,11$. In this paper, which could be seen as a completion of our preceding investigation, we find solution for the case of singular primes $p=2,11$.

Our research was inspired by the relations between the primitive periods of sequences obtained by reducing Tribonacci sequence by a given prime modulus $p$ and by its powers $p^t$, which were deduced by M. E. Waddill. In this paper we derive similar results for the case of a Tribonacci sequence that starts with an arbitrary triple of integers.

New bounds are given for the exponential sum$$\sum _{P\le p\<2P}^{}e\left(\alpha p^k\right)$$were $k\ge 5,p$ denotes a prime and $e\left(x\right)=exp\left(2\pi ix\right)$.

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously...

It is proved that for every k there exist k triples of positive integers with the same sum and the same product.

In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

We give exhaustive list of biquadratic fields $K=\mathbb{Q}(i,\sqrt{m})$ and $K=\mathbb{Q}(\sqrt{2},\sqrt{m})$ without $2$-exotic symbol, i.e. for which the $2$-rank of the Hilbert kernel (or wild kernel) is zero. Such $K=\mathbb{Q}(i,\sqrt{m})$ are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The $2$-rank of tame, regular and wild kernel of $K$-theory are connected with local and global problem of embedding in a $Z_2$-extension. Global class field theory can describe the $2$-rank of the Hilbert...

Dans cet article, nous allons démontrer qu’étant donné $G$, un sous-groupe fini de $G{l}_n\left(Z\right)$, il n’y a, à $G$-équivalence près, qu’un nombre fini de formes $G$-parfaites (resp. $G$-eutactiques, $G$-extrêmes).

We are interested whether there is a nonnegative integer $u_0$ and an infinite sequence of digits $u_1,u_2,u_3,\cdots$ in base $b$ such that the numbers $u_0{b}^n+u_1{b}^{n-1}+\cdots +u_{n-1}b+u_n,$ where $n=0,1,2,\cdots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p\in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b\in \{2,3,4,6\},$ and that no unavoidable set exists in base $b=5.$ Now, we prove...

We survey some of the universality properties of the Riemann zeta function $\zeta \left(s\right)$ and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator $𝔞$ (mapping...