Luis Baez-Duarte, Michel Balazard, Bernard Landreau, Eric Saias
A first encounter with $A(λ) := \int_0^{+\infty}\{λt\}\{t\}t^{-2}dt.$ This text contains the detailed calculations leading to the results of our paper "Etude de l'autocorrelation multiplicative de la fonction "partie fractionnaire"", to appear in the Ramanujan Journal.
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We describe the points of diff erentiability of the multiplicative autocorrelation function of the "fractional part" function. In connection with this question, we study series involving the fi rst Bernoulli function, the arithmetical function "number of divisors", and the Gauss map from the theory of continued fractions. A key role is played by a function defi ned in 1933 by J. R. Wilton, similar to the Brjuno function of dynamical systems theory. A unifying theme of our exposition is the use of functional equations involving the Gauss map, allowing us to reprove and re fine a theorem of Wilt…
We give an asymptotic formula for the quadratic total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.
We give an asymptotic formula for the total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.
Let $b > a > 0$. We prove the following asymptotic formula $$\sum_{n\ge 0} \big\lvert\{x/(n+a)\} - \{x/(n+b)\}\big\rvert = \frac{2}πζ(3/2)\sqrt{cx} + O(c^{2/9}x^{4/9}),$$ with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.
We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.
We use Möbius inversion and the Bernoulli polynomials to prove inequalities between the logarithmic summatory function of the Möbius function and weighted averages of its ordinary summatory function.
We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.
In this note, following results from Henri Cohen and Winfried Kohnen, we show that for all integer k greater than 12 and divisible by 4, there exists a cuspidal eigenform of weight k for the full modular group SL2(Z) such that its Hecke L-function does not vanish on k/2.
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3
本文研究非负乘性函数在光滑整数上的对数平均，在单侧和双侧假设下得到改进的精确估计，并据此给出了 Johnson--Selberg 素数幂筛法的有效形式。
Let $α\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^α$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.
We study, on average over f, zeros of the L-functions of primitive weight two forms of level q (fixed). We prove, on the one hand, density theorems for the zeros (similar to the results of Bombieri, Jutila, Motohashi, Selberg in the case of characters), which are applied in \cite{KM} to obtain a sharp unconditionnal estimate of the (analytic) rank of the new part of J_0(q); and, on the other hand, non-vanishing theorems at the critical point, showing that a positive proportion of L-functions are non zero there.
Let $X$ be a genus $d$ curve with $d\geq 2$ defined over a global function field $K$ of characteristic $p>0$ with $p>2d+1$. Suppose $X$ non-isotrivial. Let $Γ$ be a sub-group of $J(K_s)$, where $J$ is the jacobian of $X$ and $K_s$ is a separable closure of $K$, verifying $Γ/p Γ$ finite. Then one shows that $X\cap Γ$ has finite cardinal and one provides an explicit upper bound. This generalizes a result of Buium and Voloch.
For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}μ(d)$ where $μ$ denotes the Möbius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We show that, for a suitable, explicit, constant $L>0$ and some absolute $c>0$, we have $S(x,z)= Lx+O\left({x/{\cal L}(3ξ)^c}\right)$ uniformly for $x\geqslant 1$, $ξ\leqslant z\leqslant x/ξ$.
In this article, we give an explicit description of the invertible functions on the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. We identify them with some distribution spaces over the profinite set of $K$-rationnal points of the projective space. The strategy consists of constructing a map from these distributions to the invertible functions following the methods of Schneider-Stuhler, Iovita-Spiess, de Shalit. We show that it is compatible with the isomorphisms they constructed to compute étale and de Rham cohomology in degree $1$ and that this property forces our d…
Modular symbols for the congruence subgroup $Γ_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for $\mathbf{F}_q(T)$. The main statement also provides a key-step for a result towards the uniform…
Let $(f_1,f_2)$ a $2$-tuple of arithmetic functions and $$Δ_3(n,f_1,f_2):=\sup\limits_{\substack{(u_1,u_2) \in \mathbb{R}^{2} \\(v_1,v_2) \in [0,1]^{2}}}\Big\lvert \sum\limits_{\substack{d_1 d_{2} \mid n \\ e^{u_i}<d_i\leqslant e^{u_i+v_i}}}{f_1(d_1) f_{2}(d_{2})} \Big\rvert{\rm .}$$ In this paper, we give an upper bound of the second moment of $Δ_3(n,χ_1,χ_2)$ when $χ_1$ and $χ_2$ are two non principal Dirichlet characters, following methods developed by La Bretèche and Tenenbaum. This upper bound is a main step for the asymptotic counting of the number of ideals of norm fixed, which will be …
The Brjuno function attains a strict global minimum at the golden section.
Invariance Galoisienne des zéros centraux de fonctions L
研究证明了L函数在1/2处的消去性质及其epsilon因子的伽罗瓦不变性，通过构造离散自动形式表示和处理intertwining运算符的全纯性。
We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin's L-functions at negative integers. This generalizes conjectures by Colmez and Gross-Deligne and complements Beilinson's conjectures for the Artin motives. We announce several results in the direction of these statements.
The greatest lower bound of the real parts of the roots of a partial sum of the Dirichlet series of Riemann's zeta function is asymptotically equivalent to the opposite of the number of terms of this sum, multiplied by the Napierian logarithm of 2, when this number of terms tends to infinity.
Pureté de l'approximation forte sur le corps des fonctions d'une courbe algébrique complexe
该研究证明复代数曲线函数域上，半单代数群齐性空间及低次仿射光滑完全交的余维2闭子集补集满足去掉非空有限位集的强逼近性质。
On the rank of Jacobians over function fields.} Let $f:\mathcal{X}\to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $\mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.
Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit l…
Minima successifs des réseaux et pentes des fibrés vectoriels sur les corps de fonctions globaux
本文利用范畴等价，给出了赋范 $A$-格的 Mahler 逐次极小值与有限域上射影曲线 ${\f C}$ 上向量丛的 Harder-Narasimhan 斜率之间的精确关系。
In a previous paper, we computed the spectrum of the singular Laplacian attached to canonical metrics on $\mathbb{P}^1$. In this article, we study $ζ_\infty$, the Zeta function associated to this spectrum. We prove that it admits a meromorphic continuation to $\mathbb{C}$. Moreover, this continuation is holomorphic at zero. We compute $ζ_\infty(0)$ and $ζ'_\infty(0)$.
Finitude du groupe de Tate-Shafarevich pour les groupes de type multiplicatif constants sur des corps des fonctions
本文研究了函数域扩张上乘性类型群的Tate-Shafarevich群，证明了在满足两个技术条件时该群有限，并在整数环正则时该群平凡。
We show a weak form of the function field version of Oesterle's abc conjecture. It asserts that, if $B$ is a complex projective connected curve, the number of intersection points, counted without multiplicities, of a fixed divisor $D$ of degree $d>0$ over $B$ with the graph $H$ of a section $h:B\to B\times \bP^1$ to the first projection is at least $(d-2)n-C(B,D)$, where $n$ is the degree of $H$ over $\bP^1$, and $C(D,B)$ a constant depending only on these two data. We show this number is at least $(d-2[\sqrt {d}]).n-C(D,B)$. The constant is ineffective.
Nous étudions la nature arithmétique de $q$-analogues des valeurs $ζ(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $ζ_q(s)= \sum_{k=1} ^{\infty}q^k \sum_{d\mid k} ^{}d^{s-1}$, $s=1,2,...$, o{ù} $q$ est un nombre complexe, $| q|<1$ (ces fonctions sont intimenent liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l'espace vectoriel engendré sur $\mathbb Q$ par $1,ζ_q(3), ζ_q(5),..., ζ_q(M)$ est au moins $c_1\cdot\sqrt{M}$, …
Une remarque sur l'arborification de Matula
本文提出利用Matula的arborification方法研究自然数上莫比乌斯和利乌维函数的部分和，揭示其数论性质。
Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a …
Let $k$ be a field of characteristic 0 and let $K$ be the function field of a smooth projective geometrically integral $k$-curve $X$. Let $T$ be a $K$-torus. In this article, we aim at studying the space of adelic points $T(S,\mathbb{A}_K)$ of $T$ outside a finite set $S$ of closed points of $X$. We start by proving that the group $T(K)$ of rational points of $T$ is always discrete (hence closed) in $T(S,\mathbb{A}_K)$. We then describe the quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ in each of the following three cases: $k$ is an algebraically closed field, $k$ is the field of Laurent series $\…
Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main concept. In this article, I show that the spectral amplitude of boolean functions, which is linked to their nonlinearity, is of the order of $2^{m/2}\sqrt{m}$ in mean, whereas its range is bounded by $2^{m/2}$ and $2^m$. Moreover I examine a conjecture of Patterson and Wiedemann saying that the minimum of this spectral amplitude is as close as desired to $2^{m/…
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