In this paper, we present a new point of view on the renormalization of some exponential sums stemming from number theory. We generalize this renormalization procedure to study some matrix cocycles arising in spectral problems of quantum mechanics

For the general modulo $q\ge 3$ and a general multiplicative character $\chi$ modulo $q$, the upper bound estimate of $\left|S\right(m,n,1,\chi ,q\left)\right|$ is a very complex and difficult problem. In most cases, the Weil type bound for $\left|S\right(m,n,1,\chi ,q\left)\right|$ is valid, but there are some counterexamples. Although the value distribution of $\left|S\right(m,n,1,\chi ,q\left)\right|$ is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for $k$-th Kloosterman sums and analytic method to study the asymptotic properties...

About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let $p$ be a prime, and let $N(k;p)$ denote the number of all $1\le a_i\le p-1$ such that $a_1{a}_2\cdots a_k\equiv 1modp$ and $2\mid a_i+{\overline{a}}_i+1,$$i=1,...$

L’étude des systèmes dynamiques non archimédiens initiée par J. Lubin conduit à déterminer la ramification de séries à coefficients dans un corps fini $k$, qui commutent entre elles pour la loi $\circ$. Dans cet article nous traitons le cas des sous-groupes abéliens de $t+t^{2}k\left[\left[t\right]\right]$ qui correspondent par le foncteur corps de normes aux extensions abéliennes des extensions finies de ${\mathbb{Q}}_p$, dont la ramification se stabilise dès le début.

Nous présentons ici une étude complémentaire de notre travail en collaboration avec G. Berck et A. Bernig sur l’entropie volumique des géométries de Hilbert. Outre la présentation de nos résultats dont les démonstrations sont accessibles dans le travail susmentionné, on trouvera ici des exemples de géométrie pour lesquels le calcul de l’entropie est possible ainsi que diverses remarques quant aux conséquences de nos travaux.

Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that $$K=Q\left(\sqrt{A(D+B\sqrt{D})}\right),$$ where $$A\text{is}\text{squarefree}\text{and}\text{odd},D=B^2+C^2\text{is}\text{squarefree},B>0,C>0,GCD(A,D)=1.$$ The conductor $f\left(K\right)$ of $K$ is $f\left(K\right)=2^l\left|A\right|D$, where $$l=\left\{\begin{array}{c}3,\text{if}D\equiv 2(mod4)\text{or}D\equiv 1(mod4),B\equiv 1(mod2),\hfill \\ 2,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 3(mod4),\hfill \\ 0,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 1(mod4).\hfill \end{array}\right.$$ A simple proof of this formula for $f\left(K\right)$ is given, which uses the basic properties of quartic Gauss sums.

Recently Garashuk and Lisonek evaluated Kloosterman sums K (a) modulo 4 over a finite field F3m in the case of even K (a). They posed it as an open problem to characterize elements a in F3m for which K (a) ≡ 1 (mod4) and K (a) ≡ 3 (mod4). In this paper, we will give an answer to this problem. The result allows us to count the number of elements a in F3m belonging to each of these two classes.