Soit F ∈ Z[X1, . . . ,Xn] une forme décomposable, c’est-à-dire un polynôme homogène de degré d qui peut être factorisé en formes linéaires sur C. Notons NF (m) le nombre de solutions entières à l’inégalité |F(x)| ≤ m et VF (m) le volume de l’ensemble {x ∈ Rn :|F(x)| ≤ m}. En 2001, Thunder [19] a prouvé une conjecture de W.M. Schmidt, énonçant que, sous des conditions de finitude appropriées, on a NF (m) << m n/d où la constante implicite ne dépend que de n et d. En outre, il a montré une formule asymptotique NF (m) = m n/d V (F) + OF (m n/(d+n−2)) où, cependant, la constante implicite dépend de F. Dans des articles ultérieurs, la préoccupation de Thunder était d’obtenir une formule asymptotique similaire, mais avec la borne supérieure du terme d’erreur |NF (m) −m n/dV (F)| ne dépendant que de n et d. Dans [20] et [22], il a réussi à prouver que si gcd(n, d) = 1, la constante implicite dans le terme d’erreur peut en effet être fonction uniquement de n et d. L’objectif principal de cette thèse est d’étendre les résultats de Thunder au cadre p-adique. `A savoir, nous sommes intéressés par les solutions à l’inégalité |F(x)| · |F(x)|p1 . . . |F(x)|pr ≤ m en x = (x1, x2, . . . ,xn) ∈ Zn avec gcd(x1, x2, . . . ,xn, p1 · · · pr) = 1. (5.4.9) où p1, . . . , pr sont des nombres premiers distincts et |·|p désigne la valeur absolue p-adique habituelle. Le chapitre 1 est consacré au cadre p-adique de ce problème et aux preuves des lemmes auxiliaires. Le chapitre 2 est consacré à l’extension des résultats de Thunder de [19]. Dans le chapitre 3, nous montrons l’effectivité de la condition sous laquelle le nombre de solutions de (5.4.9) est fini. Le chapitre 4 et le chapitre 5 généralisent les résultats de Thunder dans [20], [21] et [22]. / Let F ∈ Z[X1, . . . ,Xn] be a decomposable form, that is, a homogeneous polynomial of degree d which can be factored into linear forms over C. Denote by NF (m) the number of integer solutions to the inequality |F(x)| ≤ m and by VF (m) the volume of the set{x ∈ Rn : |F(x)| ≤ m}. In 2001, Thunder [19] proved a conjecture of W.M. Schmidt, stating that, under suitable finiteness conditions, one has NF (m) << mn/d where the implicit constant depends only on n and d. Further, he showed an asymptotic formula NF (m) = mn/dV (F) + OF (mn/(d+n−2)) where, however, the implicit constant depends on F. In subsequent papers, Thunder’s concern was to obtain a similar asymptotic formula, but with the upper bound of the error term |NF (m)−mn/dV (F)| depending only on n and d. In [20] and [22], hemanaged to prove that if gcd(n, d) = 1, the implicit constant in the error term can indeed be made depending only on n and d.The main objective of this thesis is to extend Thunder’s results to the p-adic setting. Namely, we are interested in solutions to the inequality |F(x)| · |F(x)|p1 . . . |F(x)|pr ≤ m in x = (x1, x2, . . . ,xn) ∈ Zn with gcd(x1, x2, . . . ,xn, p1 · · · pr) = 1. (5.4.3)where p1, . . . , pr are distinct primes and | · |p denotes the usual p-adic absolute value.Chapter 1 is devoted to the p-adic set-up of this problem and to the proofs of the auxiliary lemmas. Chapter 2 is devoted to extending Thunder’s results from [19]. In chapter 3, we show the effectivity of the condition under which the number of solutions of (5.4.3) is finite. Chapter 4 and chapter 5 generalize Thunder’s results from [20], [21] and [22].

The lattice point problem in dynamical systems investigates the distribution of certain objects with some length property in the space that the dynamics is defined. This problem in different contexts can be interpreted differently. In the context of symbolic dynamical systems, we are trying to investigate the growth of N(T), the number of finite words subject to a specific ergodic length T, as T tends to infinity. This problem has been investigated by Pollicott and Urbański to a great extent. We try to investigate it further, by relaxing a condition in the context of deterministic dynamical systems. Moreover, we investigate this problem in the context of random dynamical systems. The method for us is considering the Fourier-Stieltjes transform of N(T) and expressing it via a Poincaré series for which the spectral gap property of the transfer operator, enables us to apply some appropriate Tauberian theorems to understand asymptotic growth of N(T). For counting in the random dynamics, we use some results from probability theory.

The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).