We give a necessary and sufficient condition for the following property of an integer $d\in\mathbb N$ and a pair $(a,A)\in\mathbb R^2$: There exist $κ> 0$ and $Q_0\in\mathbb N$ such that for all $\mathbf x\in \mathbb R^d$ and $Q\geq Q_0$, there exists $\mathbf p/q\in\mathbb Q^d$ such that $1\leq q\leq Q$ and $\|\mathbf x - \mathbf p/q\| \leq κq^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $κ= Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if $\mathbb R^d$ is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case $d = 1$ we describe the set of exceptions in terms of classical Diophantine conditions.
Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative proof of it based on a previous result of us. Also we get an estimation of the prime counting function in the special cases.
We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime number theory.
We give new characterizations of the Midy's property and using these results we obtain a new proof of a special case of the Dirichlet's theorem about primes in arithmetic progression.
If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting.
We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many polynomials c(X)\in F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.
We show that the module of integral points on a Drinfeld module satisfies a an analogue of Dirichlet's unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated sub-module of the module of integral points. We use the results to give a precise formulation of a conjectural analogue of the class number formula.
It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<σ<1$; this is certainly not true for $σ>1$. Answering a question of Bombieri and Ghosh, we give a simple characterization of the analytic functions approximable by translates of $L$-functions in the half-plane of absolute convergence. Actually, this is a special case of a general rigidity theorem for translates of Dirichlet series in the half-plane of uniform convergence. Our results are closely related to Bohr's equivalence theorem.
We generalize Dirichlet's $S$-unit theorem from the usual group of $S$-units of a number field $K$ to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over $S$. Specifically, we demonstrate that the group of algebraic $S$-units modulo torsion is a $\bQ$-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over $\mathbb{Q}$ retain their linear independence over $\mathbb{R}$.
For a given Dirichlet character $χ(n) = e^{i θ_n}$, we prove central limit theorems for the series $\sum_{p'} \cos θ_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$ are integers based on a variant of Cramér's random model for the primes. For non-principal characters, we use these results to show that the Generalized Riemann Hypothesis for the associated $L$-function is true with probability equal to one. For principal characters we propose how to extend these arguments to $\Re (s) = t \to \infty$.
If $a$ and $d$ are relatively prime, we refer to the set of integers congruent to $a$ mod $d$ as an `eligible' arithmetic progression. A theorem of Dirichlet says that every eligible arithmetic progression contains infinitely many primes; the theorem follows from the assertion that every eligible arithmetic progression contains at least one prime. The Jacobsthal function $g(n)$ is defined as the smallest positive integer such that every sequence of $g(n)$ consecutive integers contains an integer relatively prime to $n$. In this paper, we show by a combinatorial argument that every eligible ari…
Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].
In this article, we study the logarithm of the central value $L\left(\frac{1}{2}, χ_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $δ$ over a fixed finite field $\mathbb{F}_q$ in the limit as $δ\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, χ_D\right)\big|$ is asymptotically bounded above by the Gaussian distribution of mean $\frac{1}{2}\log °(D)$ and variance $\log °(D)$. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian …
We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet's theorem in Diophantine approximation and prove very general results in this context.
In this paper, we would like to propose a fundamental question about a higher dimensional analogue of Dirichlet's unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.
It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime number counting function to provide a new proof of Dirichlet's theorem. We show that if $a\in \mathbb{N},k\in \mathbb{N},a_{k}=(a,a+1,...,a+k-1) $ and $A=\left\{ p_{1},p_{2},...,p_{n}\right\} $, a finite set of primes. Then the number of components of $a_{k}$ that are divisible by some prime in $A$ is less than or equal to $$ \sum\limits_{\substack{ d|P(A)\\ d>1}…
The purpose of this paper is to generalize our earlier work on the logarithm of the Riemann zeta-function to linear combinations of logarithms of primitive Dirichlet $L$-functions with constant real coefficients. Under the assumption of suitable hypotheses, we prove that as $T\to \infty $, a sequence of the form $(a_1\log|L(ρ,χ_n)|+\dots+a_1\log|L(ρ,χ_n)|)$ has an approximate Gaussian distribution with mean $0$ and variance $ \tfrac{1}{2}\big({a_1}^2+\dots+{a_n}^2 \big)\log\log T$. Here $a_1, \dots, a_n \in \mathbb{R}$, each of the $χ_i$ is a primitive Dirichlet character modulo $M_i$ with $M_…
We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to additive problems with prime variables. In particular, we are able to improve on a recent result of J.Y. Liu and K.M. Tsang on the size of the solutions of linear equations with prime variables.
We study the distribution of closed geodesics for the modular surface. We improve the error term in the prime geodesic theorem, and obtain results on prime geodesics in very short intervals conditionally on the generalized Riemann Hypothesis for Dirichlet L-functions. We emphasize a connection between the closed geodesics and certain Dirichlet L-functions.
In this paper we prove a function field version of a theorem by Rudnick and Soundararajan about lower bounds for moments of quadratic Dirichlet $L$-functions. We establish lower bounds for the moments of quadratic Dirichlet $L$--functions associated to hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_{q}$ in the large genus $g$ limit.
We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces.
We give an integrability condition on a function $ψ$ guaranteeing that for almost all (or almost no) $x\in\mathbb{R}$, the system $|qx-p|\leq ψ(t)$, $|q|<t$ is solvable in $p\in \mathbb{Z}$, $q\in \mathbb{Z}\smallsetminus \{0\}$ for sufficiently large $t$. Along the way, we characterize such $x$ in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.
We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group $Γ_0(13)$. The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.
We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization technique…
Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $Γ(1)$. Weil extended this result to $Γ_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. Conrey and Farmer extended Hecke's result for certain small $N$, assuming that the local factors in the Euler product of $F(s)$ were of a special form. We make the same assumption on the Euler product and describe an approac…
In this paper, we study an analytic curve $φ: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $φ$ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into some homogeneous space $G/Γ$, and prove that under the action of some expanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates of the curve tend to be equidistributed in $G/Γ$, as $t \rightarrow +\infty$.
We show that for any $ε<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $μ$ there is $ε_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $ε<ε_0$, and for $μ$-almost every $Y$, Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals. Our results extend those …
Let $ψ$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$ one can find $\boldsymbol{p} \in \mathbb{Z}^m$, $\boldsymbol{q} \in \mathbb{Z}^n\smallsetminus\{\boldsymbol{0}\}$ satisfying $\|A\boldsymbol{q}-\boldsymbol{p}\|^m< ψ({t})$ and $\|\boldsymbol{q}\|^n<{t}$. This property was introduced by Kleinbock and Wadleigh in 2018, generalizing the property of $A$ being Dirichlet improvable which dates back to Davenport and Schmidt (1969). In the present paper, we gi…
We give an integrability criterion on a real-valued non-increasing function $ψ$ guaranteeing that for almost all (or almost no) pairs $(A, \textbf{b})$, where $A$ is a real $m\times n$ matrix and $\textbf{b} \in \mathbb{R}^m$, the system $\|A \textbf{q}+\textbf{b}-\textbf{p}\|^m< ψ({T})$, $\|\textbf{q}\|^n<{T}$ is solvable in $\textbf{p} \in \mathbb{Z}^m$, $\textbf{q} \in \mathbb{Z}^n$ for all sufficiently large $T$. The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose …
Let $X\cong \operatorname{SL}_2(\mathbb R)/\operatorname{SL}_2(\mathbb Z)$ be the space of unimodular lattices in $\mathbb R^2$, and for any $r\ge 0$ denote by $K_r\subset X$ the set of lattices such that all its nonzero vectors have supremum norm at least $e^{-r}$. These are compact nested subset{s} of $X$, with $K_0 = {\bigcap}_{r}K_r$ being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in $\mathbb R^2$ centered at the origin to derive an asymptotic formula for the volume of sets $K_r$ as $r\to 0$. …
Limit theorems for random Dirichlet series with summation over primes, with an application to Rademacher random multiplicative functions
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv（基辅塔拉斯舍甫琴科国立大学计算机科学与控制论学院）
