Search

We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al. under Generalized Riemann Hypothesis for automorphic L-function.

RésuméUne généralisation du théorème de Beurling et Nyman établit que l'hypothèse de Riemann pour une fonction F de la classe de Selberg est équivalente à l'appartenance de la fonction χ, indicatrice de l'intervalle ]0,1[ à l'adhérence d'un sous-espace de fonctions BF dans l'espace L2(0,+∞). Dans cet article, l'auteur étend aux fonctions F de la classe de Selberg un résultat de Báez-Duarte en donnant une construction d'une suite de BF qui, sous l'hypothèse de Riemann pour la fonction F, converge dans L2(0,+∞) vers la fonction χ.AbstractA generalisation of the Beurling and Nyman criterion implies that the Riemann hypothesis for a function F in the Selberg class is equivalent to the statement that χ, the characteristic function of (0,1) belongs to the adherence in L2(0,+∞) of a subspace BF. In this article, the author generalises a result of Báez-Duarte to the Selberg class and gives a construction of some sequence of BF converging to χ in L2(0,+∞) under Riemann hypothesis for F.

In the study of substitutative dynamical systems and Pisot number systems, an algebraic condition, which we call ‘weak finiteness’, plays a fundamental role. It is expected that all Pisot numbers would have this property. In this paper, we prove some basic facts about ‘weak finiteness’. We show that this property is valid for cubic Pisot units and for Pisot numbers of higher degree under a dominant condition.

International audience; We are interested in the analogue of a result proved in the number field case by E. Brown and C.J. Parry and in the function field case in odd characteristic by Zhang Xianke. Precisely, we study the ideal class number one problem for imaginary quartic Galois extensions of $k=\fq(x)$ of Galois group $\integers /2\integers \times\integers /2\integers$ in even characteristic.\\Let $L/k$ be such an extension and let $K_1$, $K_2$ and $K_3$ be the distinct subfields extensions of $L/k$. In even characteristic, the fields $K_i$ are Artin-Schreier extensions of $k$ and $L$ is the compositum of any two of them.\\Using the factorization of the zeta functions of this fields, we get a formula between their ideal class numbers which enables us to find all imaginary quartic Galois extensions $L/k$ of Galois group $\integers /2\integers \times\integers /2\integers$ with ideal class number one.