Richard Carr, Cormac O'Sullivan
A set of real $n$th roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.
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We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick--Waxman and Keating--Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on…
Quantitative linear independence for square roots
本文用概率方法证明平方根整数线性组合的定量下界，改进了经典乘积界对项数K的依赖。
Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\cos (πr_1)$, and $\cos (πr_2)$ to be rationally independent. Extending a classical fact sometimes attributed to I. Niven, the result even yields linear independence over larger number fields. The tools employed in the proof are applicable also in the case of more than two trigonometric numbers. As an application, a complete classification is given of all planar triangles with rational angles and side lengths each containing at most one square root. Such a classificat…
We prove linear independence results for values of (a certain class of) q-hypergeometric series in a quantitative form.
Inspired by Warren Sinnott 's method we prove a linear independence result modulo p for the Iwasawa power series associated to Kubota-Leopoldt p-adic L-functions.
We prove, in a quantitative form, linear independence results for values of a certain class of q-series, which generalize classical q-hypergeometric series. These results refine our recent estimates.
We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.
Linear independence of periods related to polylogarithms
本文首次提出了多项级数值在代数数域上线性独立性的标准，特别地，得出了在代数数域上不同点处多项级数乘积的线性独立性新结果，方法基于针对多项级数的Padé型近似显式构造。
In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.
We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.
We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear independence relations over the field of algebraic functions are consequences of the known relations of Anderson and Thakur arising from the functional equations of the Gamma function.
Linear forms in logarithms over connected commutative algebraic groups over the algebraic numbers field have been studied widely. However, the theory of linear forms in logarithms over noncommutative algebraic groups have not been developed as the one of the commutative algebraic groups and in this paper we start studying linear forms in logarithms over affine groups.
We prove linear independence of indefinite iterated Eisenstein integrals over the fraction field of the ring of formal power series $\mathbb{Z}[[q]]$. Our proof relies on a general criterium for linear independence of iterated integrals, which has been established by Deneufch{â}tel, Duchamp, Minh and Solomon. As a corollary, we obtain $\mathbb{C}$-linear independence of indefinite iterated Eisenstein integrals, which has applications to the study of elliptic multiple zeta values, as defined by Enriquez.
In 1985, Yu. V. Nesterenko produced a criterion for linear independence, which is a variant of Siegel's. While Siegel uses upper bounds on full systems of forms, Nesterenko uses upper and lower bounds on sufficiently dense sequences of individual forms. The proof of Nesterenko's criterion was simplified by F. Amoroso and P. Colmez in 2003. More recently, S. Fischler and W. Zudilin produced a refinement, together with a much simpler proof. This new proof rests on a simple argument which we expand here. We get a new result, which contains Nesterenko's criterion, as well as criteria for algebraic…
Rodrigues formula and linear independence for values of hypergeometric functions with parameters vary
本文通过证明广义 Rodrigues 公式，构造含 Gauss 超几何函数在内的 Padé 逼近，从而建立参数可变时其函数值在数域上的新线性无关性准则。
The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.
Linear independence of values of the $q$-exponential and related functions
本文证明当$q$为Pisot-Vijayraghavan数时，$q$-指数函数及其导数在代数点上的值线性无关，并推广至相关函数。
In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated by a linear recurrence. In order to prove this result, we reduce the algebraic independency to that of Mahler functions of several variables by shifting the linear recurrence and apply the theory of Mahler functions.
Linear independence of theta series of positive-definite even unimodular lattices
研究正定单模偶格中theta级数的线性独立性，确定其最小度数范围。
The linear independence of $1$, $ζ(3)$, and $ζ(4)$
本文结合 Fauzan 行列式方法与经典同时逼近，证明了 $1$、$\zeta(3)$、$\zeta(4)$ 在有理数域上线性无关。
We show that two distinct singular moduli $j(τ),j(τ')$, such that for some positive integers $m, n$ the numbers $1,j(τ)^m$ and $j(τ')^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most $2$. This completes a result of Riffaut, who proved the above theorem except for two explicit pair of exceptions consisting of numbers of degree $3$. The purpose of this article is to treat these two remaining cases.
On the linear independence of $p$-adic polygamma values
本文提出$p$-进多伽马函数值的新线性无关判别准则，推导$p$-进Hurwitz zeta函数多族平移值的线性无关性，改进推广已有结果，核心是利用正交多项式Rodrigues公式的差分类似构造第二类Padé型逼近。
In this paper, we study transcendence theory for Thakur multizeta values in positive characteristic. We prove an analogue of the strong form of Goncharov's conjecture. We also establish the same result for Carlitz multiple polylogarithms at nontrivial algebraic points.
We consider the problem of simultaneous approximation of real numbers $θ_1, \ldots,θ_n$ with rationals and the dual problem of approximating zero with the values of the linear form $x_0+θ_1x_1+\ldots+θ_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation, which proves their result. We also derive several corollaries previously unknown. Particularly, we show that, together with the transference inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalit…
A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers contain infinitely many irrational numbers. This result is proved by T.Rivoal if the function considered is the Riemann zeta function, and this paper gives its generalization to more general Dirichlet series.
A linear independence criterion for $p$-adic Hurwitz zeta values at odd integers
该研究推广F. Beukers的定理，建立p进Hurwitz zeta函数在奇数s≥3处取值的线性独立性新准则，通过构造满足f(z)=f(1-z)的Laurent级数的Padé逼近完成证明，补充了p进多伽马值线性独立性的相关工作。
In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small at $k$ points. This is a generalization to vectors of Nesterenko's linear independence criterion (which corresponds to $k=1$), used by Ball-Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski's theorem on convex bodies.
Linear independence of continued fractions with algebraic terms
本文给出了正代数数序列和数域的条件，以确保由连分数定义的数与1在该数域上线性无关。
Modular forms with non-vanishing central values and linear independence of Fourier coefficients
本文通过将VanderKam定理推广到高权模符号，证明对大素数p及D²≪p，Hecke算子T₁,…,T_D在权2k尖点模符号空间的winding元素上线性无关，从而给出中心临界值非零的新形式数量界及其模l约化的线性无关性。
Possible transcendental nature of Euler's constant $γ$ has been the focus of study for sometime now. One possible approach is to consider $γ$ not in isolation, but as an element of the infinite family of generalised Euler-Briggs constants. In a recent work \cite{GSS}, it is shown that the infinite list of generalized Euler-Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler-Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see \thmref{pre} and \thmref{linear-ind}) on …
Independence of the Diophantine exponents associated with linear subspaces
本文研究Schmidt提出的丢番图逼近推广问题，证明线性子空间丢番图指数相关函数间无光滑关系，并构造可计算指数的子空间。
Let $m\ge 2$ be an integer, $K$ an algebraic number field and $α\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in $1,{\rm{log}}(1+α),\ldots,{\rm{log}}^{m-1}(1+α)$ with algebraic integer coefficients in both complex and $p$-adic cases (see Theorem $2.1$ and Theorem $2.4$). Especially, in the complex case, our result is a refinement of the result of Nesterenko-Waldschmidt on the lower bound of linear form in certain values of power of logarithms. The main integrant is based on Hermite-Mahler's Padé approximation of e…
Linear independence of values of hypergeometric functions and arithmetic Gevrey series
证明广义超几何函数${}_pF_q$在不同代数点处值的线性独立性，利用II型帕德逼近统一构造及新型非零论证，推广加强前人成果。
Linear independence of values of Dirichlet $L$ functions
本文研究了Dirichlet L函数在特定条件下的线性无关性，通过改进的Siegel引理和推广的线性无关判据，给出了其值的维数下界，并推广了Fischler的先前结果。
The threshold for linear independence of multiple zeta values in positive characteristic
本文证明了在正特征下，固定重量的多ζ值在$\mathbb{F}_q$上线性独立，直到重量2q，而重量2q+1时存在唯一的线性关系，推翻了Thakur的猜想。
A $v$-adic variant of Anderson-Brownawell-Papanikolas linear independence criterion and its application
本文研究了v-adic卡利茨多重polylogarithms在代数点上的线性关系，并证明了当v的度数为一时，这些关系可以由k-线性关系产生，同时提出了一个函数域版本的furusho-yamashita猜想。
The linear independence of $1$, $ζ(2)$, and $L(2,χ_{-3})$
本研究提出一种新型算术和乐界方法，应用于Zagier构造，证明了经典Dirichlet L值$L(2,\chi_{-3})$的无理性，还确立了$1$、$\zeta(2)$与$L(2,\chi_{-3})$的$\mathbf{Q}$-线性无关性，并给出方法在其他无理性问题中的应用。
Let $(i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}$ and $S_{i,j}$ be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over $\mathbb{Q}$ of the numbers \[ 1, \quad \sum_{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}, \] where $b\geq2$ is an integer and $a_{i,j}(n)$ are bounded nonzero integer-valued functions on $S_{i,j}$. Moreover, we also establish a necessary and sufficient condition on the subset $\mathcal{A}$ of $\mathbb{N}\times \mathbb{N}_{\ge…
We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the $\mathbb{Q}$-vector space spanned by $1$ and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite-Padé approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are rela…
Linear independence of periods for the symmetric square $L$-functions
本文研究完全模群尖点形式空间上对称平方 $L$-函数周期的线性无关性，证明当权充分大时任意固定数量周期线性无关，并给出显式数量界限。
A linear independence criterion for certain infinite series with polynomial orders
本文研究了当 $q$ 为 Pisot 或 Salem 数时，特定无穷级数在域 $\mathbb{Q}(q)$ 上的线性无关性，并针对 $a_j(n)$ 为多项式的情形提出了线性无关性判据。
The study of linear independence of $L(k, χ)$ for a fixed integer $k>1$ and varying $χ$ depends critically on the parity of $k$ vis-à-vis $χ$. This has been investigated by a number of authors for Dirichlet characters $χ$ of a fixed modulus and having the same parity as $k$.The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a s…
An effective Linear Independence conjecture for the zeros of the Riemann zeta function and applications
本文基于对数线性型理论中的启发式论证，提出黎曼zeta函数零点纵坐标的有效线性无关性猜想，并在此猜想及Gonek-Hejhal猜想下，得到素数定理误差项和Möbius函数求和函数的Omega结果，条件性解决Gonek部分猜想。
The Linear Independence hypothesis (LI), which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as was pointed out by Rubinstein and Sarnak. In this paper, we prove that a function field analogue of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite …
Algebraic independence of infinite series
本文给出了有理数级数集合代数独立的充分条件，并由此推导出级数无理性与有理数线性独立的新结果。
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a new technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Section 2 and 3, using the technique together with Mahler's method, we can reduce the al…
We introduce and study algebraic dynamical systems generated by triangular systems of rational functions. We obtain several results about the degree growth and linear independence of iterates as well as about possible lengths of trajectories generated by such dynamical systems over finite fields. Some of these results are generalisations of those known in the polynomial case, some are new even in this case.
Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of "most" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be seen as an analogue of conjectures of linear independence among ordinates of zeros of L-functions over number fields. As a corollary, we find, for "most" pairs of distinct algebraic curves over a finite field, the limiting distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The meth…
The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $χ$ and a number field $K$ the formula $\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,χω^{1-i}) K )\geq \frac{(1-ε)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function.
We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the $G$-function satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like algebraic independence…
Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ …
Linear independence measures for Chowla--Selberg periods
通过同时Padé近似估计超几何函数，研究特定d和D下1、π√d、Ω_D/π和π/Ω_D的线性形式，揭示其线性独立性。
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