Michael Coons
We give an upper bound for the zero order of the difference between a Mahler function and an algebraic function. This complements estimates of Nesterenko, Nishioka, and Töpfer, among others, who considered polynomials evaluated at Mahler functions.
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In this work, we are dealing with some properties relating the zeros of a polynomial and its Mahler measure. We provide estimates on the number of real zeros of a polynomial, lower bounds on the distance between the zeros of a polynomial and non-zeros located on the unit circle and a lower bound on the number of zeros of a polynomial in the disk $\{\vert x-1\vert<1\}$.
In this note we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.
We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd--Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true.
In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Padé approximations of the functions under consideration, then apply the functional equations to get an infinite sequence of approximations and use the numerical approximations obtained from this sequence.
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.
On a Problem of Mahler Concerning the Approximation of Exponentials and Logarithms
本文首先给出了代数数的丢番图逼近的两个几何估计。接下来，我们讨论的艺术状态，我们给这个话题上的进一步的部分结果。
Transcendence of multivariate Mahler functions and algebraic relations between their values
研究多元马勒函数的函数性质，证明其在特定区域的解析性与亚纯性及有理 - 超越二分法，加强相关提升定理，能计算函数值间代数关系，还证明了正则点情况下的多元下降定理。
We establish a Mahler-type estimate of weighted Fekete sums on the Berkovich projective line over an algebraically closed field of possibly positive characteristic that is complete with respect to a non-trivial and possibly non-archimedean absolute value.
The distribution of Mahler's measures of reciprocal polynomials
本文研究了复系数有界偶数次互反多项式的Mahler测度的分布。我们发现，与Mahler测度有关的分布函数在[1，\infty）上是一个互反（或反互反）Laurent多项式，在[0，1）上是恒零的.此外，这个劳伦多项式的系数是有理数乘以π的幂。我们是通过计算分布函数的梅林变换得出这一发现的。这种梅林变换是一个偶（或奇）有理函数，其极点位于小整数处，残差是有理数乘以π的幂。我们还使用这种梅林变换表明，一组倒数多项式的体积与复系数，有界度和马勒的措施小于或等于一个是一个有理数倍的幂π。
A height gap theorem for coefficients of Mahler functions
本文研究了马勒函数系数的高度增长行为，证明了存在五种不同的增长模式，并揭示了其中的间隙现象，同时推导了k-马勒函数的正则性和自动性条件。
The asymptotic Mahler measure of Gaussian periods
构造具有特别小Mahler测度的分圆整数（高斯周期）序列，研究其Mahler测度作为导子函数的渐近行为，发现增长率是维数递增的log Calabi-Yau簇族的多元Mahler测度。
Let $\mathbb{K}$ be a function field of characteristic $p>0$. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb{K}(z)$. This paper is dedicated to proving the following refinement of this theorem. Let $f_{1}(z),\ldots f_{n}(z)$ be $d$-Mahler functions such that $\overline{\mathbb{K}}(z)\left(f_{1}(z),\ldots, f_{n}(z)\right)$ is a regular extension over $\overline{\mathbb{K}}(z)$. Then, every homogeneous algebraic relation over $\overline{\mathbb{K}}$ between their values at a regular algebraic point arises as the specialization of…
Let $K$ be a field of characteristic zero and $k$ and $l$ be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten during the Eighties: a power series $F(z)\in K[[z]]$ satisfies both a $k$- and a $l$-Mahler type functional equation if and only if it is a rational function.
Power-saving bounds for Thue--Mahler and Mordell equations
该研究针对三次Thue--Mahler方程证明了对regulator具节能依赖的有效界，将Mordell方程整数解的Stark界指数降至1/2+ε，改进了相关分离下界并在Frey高度猜想的导体方面取得节能进展。
Metric Heights on an Abelian Group
作者研究了具有三角不等式的 Mahler measure 变体 $m_t(α)$，将其通用构造推广至任意 Abelian group 上的抽象函数，证明了已知结果的类似定理，并解决了先前关于 $m_t(α)$ 的开放问题。
In a recent paper the team of Cogdell, Jorgenson and Smajlović develop infinite series representations for the logarithmic Mahler measure of a complex linear form, with 4 or more variables. We establish the case of 3 variables, by bounding an integral with integrand involving the random walk probability density $a\displaystyle\int_0^\infty tJ_0(at) \displaystyle\prod_{m=0}^2 J_0(r_m t)dt$, where $J_0$ is the order zero Bessel function of the first kind, and $a$ and {$r_m$} are positive real numbers. To facilitate our proof we develop an alternative description of the integral's asymptotic beha…
Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
该研究否定解决了 Mahler 关于存在将 Liouville 数映射到 Liouville 数的超越整函数的问题，证明实解析函数若具此性质必为有理函数，并通过双高度计数估计给出定量局部结果。
The primary goal of this paper is to provide a general multiplicity estimate. Our main theorem allows to reduce a proof of multiplicity lemma to the study of ideals stable under some appropriate transformation of a polynomial ring. In particular, this result leads to a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. On the other hand, it allows to establish an improvement of Nesterenko's conditional result on solutions of systems of differential equations. We also deduce, under some condition on stable varieties, the optimal multiplicity e…
Determinants of Mahler measures and special values of $L$-functions
研究二元多项式族\(P_t\)和\(Q_t\)的马勒测度，推导其在定义CM椭圆曲线时马勒测度的公式，分类特定数域中\(t\)值，证明对应椭圆曲线相关矩阵行列式与\(L\)值的关系。
Using the theory of o-minimality we show that the $p$-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if $f_1,...,f_n$ is a finite sequence of real analytic functions $f_i:(-1,1) \to (-1,1)$ for which $f_i(0) = 0$ and $|f_i'(0)| \leq 1$ (possibly zero), $a = (a_1,...,a_n)$ is an $n$-tuple of real numbers close enough to the origin and $H(x_1,...,x_n)$ is a real analytic function of $n$ variables, then the set $\{m \in {\mathbb N} : H (f_1^{\circ m} (a_1),...…
Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization
本文给出正特征环上Dong与Shafrir结果的更简短证明，并将Skolem-Mahler-Lech定理推广至多维线性递推序列，为相关问题提供判定方法。
The range of multiplicative functions on C[x], R[x] and Z[x]
Mahler's measure is generalized to create the class of { \it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane . Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler} , we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C [x] and \R [x]. These { \it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithme…
The height of the mixed sparse resultant
我们提出了一个混合稀疏结式的高度的上界，定义为它的系数的最大模的对数。我们得到了类似的估计其马勒措施。
We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance within the context of Mahler's method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence our result provides, under some conditions, the measure of algeb…
Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that the set of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a …
Non degenerate Birkhoff functions
本文针对流形上极好逼近数零测度集中非退化条件难以检验的问题，提出一种自动满足该条件的设定，通过研究具有更好性质的对应关系替代函数来加以解决。
We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.
New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach
本文通过创建解析数论指数数据库（ANTEDB）并系统化优化指数间关系，获得了四个新指数对及Riemann zeta函数的零点密度与加性能量新估计，并支持将新指数界转化为其他相关指数的优化界。
In this paper we obtain some new estimates for the number of large values of Dirichlet polynomials. Our results imply new zero density estimates for the Riemann zeta function which give a small improvement on results of Bourgain and Jutila.
A note on zero density results implying large value estimates for Dirichlet polynomials
本文证明黎曼 zeta 函数的零密度估计可蕴含 Dirichlet 多项式的大值估计，填补了反向蕴含关系的空白。
We generalize a method of Conrey and Ghosh (Invent. Math. 94 (1988)) to prove quantitative estimates for simple zeros of modular form L-functions of arbitrary conductor.
Let $f$ be a holomorphic cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper, we shall give the relation of the number of zeros of $L_f(s)$ and the derivatives of $L_f(s)$ using Berndt's method, and an estimate of zero-density of the derivatives of $L_f(s)$ based on Littlewood's method.
Half-isolated zeros and zero-density estimates
本文提出一种对零点垂直分布敏感的新方法，证明半孤立零点很少，并在非平凡零点限于有限条固定垂直直线的假设下改进 Ingham-Huxley 零点密度估计，从而为短区间素数分布带来新推论。
A zero density estimate and fractional imaginary parts of zeros for $\mathrm{GL}_2$ $L$-functions
该研究将Selberg关于黎曼ζ函数的零点密度估计推广至任意$\u200bmathrm{GL}_2$\u200b $L$-函数，并以此分析其非平凡零点虚部与固定实向量乘积的分数部分分布特性。
Let $β'+iγ'$ be a zero of $ζ'(s)$. In \cite{GYi} Garaev and Yıldırım proved that there is a zero $β+iγ$ of $ζ(s)$ with $ γ'-γ\ll \sqrt{|β'-1/2|} $. Assuming RH, we improve this bound by saving a factor $\sqrt{\log\logγ'}$.
Explicit zero density estimate near unity
该研究首次给出ζ函数特定形式的显式零点密度估计，并将其中的常数C改进为10393/900即约11.547。
An explicit log-free zero density estimate for the Riemann zeta-function
本文为 Riemann zeta 函数给出显式无对数零点密度估计，在特定参数范围内成为已知最精确结果。
Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $χ$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $χ$, regarding the zeros of the Hecke L-function $L(s,χ)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estimates.
We improve the results from [EBS10] about the Eisenstein series $E_2(z)=1-24\sum_{n=1}^\infty \frac{nq^n}{1-q^n}$. In particular we show that there exists exactly one (simple) zero in each Ford circle and give an approximation to its location.
Considering the family of $L$-functions $\{L(s,f)\}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp forms for $SL_2(\mathbb{Z})$, we prove a zero density estimate near the central point, valid as the weight $k \to \infty$. This is an ingredient in the author's related paper, which gives an unconditional upper bound on the distribution of the central values.
We prove a log-free zero density estimate for automorphic $L$-functions defined over a number field $k$. This work generalizes and sharpens the method of pseudo-characters and the large sieve used earlier by Kowalski and Michel. As applications, we demonstrate for a particular family of number fields of degree $n$ over $k$ (for any $n$) that an effective Chebotarev density theorem and a bound on $\ell$-torsion in class groups hold for almost all fields in the family.
We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients in the vertical strip $ σ_1 < \Re s < σ_2 $, where $ 1/2 < σ_1 < σ_2 < 1 $. When the class number of the quadratic form is bigger than 1, Voronin gives a lower bound and Lee gives an asymptotic formula for the number of zeros. In this paper, we improve their results by providing a new upper bound for the error term.
In 1946, A. Selberg proved $N(σ,T) \ll T^{1-\frac{1}{4} \left(σ-\frac{1}{2}\right)} \log{T}$ where $N(σ,T)$ is the number of nontrivial zeros $ρ$ of the Riemann zeta-function with $\Re\{ρ\}>σ$ and $0<\Im\{ρ\}\leq T$. We provide an explicit version of this estimate, together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta-function on the critical line.
Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. In this paper, we prove similar results related to the first derivative of the Dirichlet $L$-functions associated with primi…
Let $χ$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,χ)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,χ)$ is first related to the distribution of zeros of a family of Dirichlet $L$-functions in a certain region, and some results on the gaps between consecutive zeros are derived. Then, by evaluating certain discrete means of the large sieve type, a contradiction can be obtained if $L(1,χ)$ is too small.
On zero-density estimates for Beurling zeta functions
本文针对满足特定整数分布的Beurling广义数系对应的Beurling zeta函数给出零点密度估计，还将结果推广到更广泛的广义Dirichlet级数，探讨了条件改进及最优性问题，推进了Beurling zeta函数的零点分布研究。
Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory classical functions which are closely linked to zeta zeroes like psi(x), theta(x), pi(x) or the k-th prime number.
A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms
研究通过建立 $\rm SL(3,\mathbb{Z})$ Hecke–Maass 尖点形式对应 $L$-函数的扭曲二阶矩渐近公式，得到其谱方面的加权零点密度估计。
A decades-long breakthrough in zero-density estimates and primes in short intervals
研究黎曼假设下零密度定理，2024年Larry Guth和James Maynard宣布新定理，加强了Ingham之前工作，本文将其置于零密度定理历史背景下并探讨对短区间素数分布的影响。
Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. Let $f \in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the distribution of zeros of $L_f(s)$ in the strip $σ\leq \Re s \leq 1$ for fixed $σ>1/2$ with respect to the imaginary part. We study estimates of \[ N_f(σ,T) = #\{ρ\in\mathbb{C} \mid L_f(ρ)=0, σ leq \Reρ\leq 1, 0 \leq \Imρ\leq T} \] for $1/2 \leq σ\leq1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivić's method.
Conditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros
在广义黎曼假设下，利用Beurling-Selberg极值函数对Dirichlet L-函数的论点均值和均方进行界估，并应用于低处零点结果的证明及获得新的零点比例下界。
A zero density estimate for Dedekind zeta functions
该研究证明了与固定伽罗瓦群的伽罗瓦扩张相关的Dedekind zeta函数族的首个不依赖强Artin猜想的零点密度估计，并将其用于移除相关数论研究中的强Artin猜想假设。
On zero-density estimates and the PNT in short intervals for Beurling generalized numbers
本文研究了整数分布为 $N(x) = Ax + O(x^θ)$ 的 Beurling 广义素数系统的 zeta 函数零点分布，通过将 Dirichlet 多项式经典均值定理推广至广义情形，显著改进了零点密度估计，并探讨了其对短区间素数定理的影响。
Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros
本文基于Selberg的结果，研究狄利克雷L函数的论估计，并证明了足够大的素数q对应的L函数最低非平凡零点高度小于1075·2π/log q，同时给出了首个零点位置的下界。
Zero density estimates for $\mathrm{GL}_2$ automorphic $L$-functions
本文证明 GL2 自守 L-函数的零密度估计，通过推广 Guth–Maynard 方法至弱 ℓ^q 界 Dirichlet 多项式，得到 N_π(σ,T) ≪ T^{5/2(1-σ)+o(1)}。
Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region. Theorem 2 gives an asymptotic estimate for the number of nontrivial zeros to height T. Theorem 3 is a zero density estimate.
An explicit form of Ingham's zero density estimate
本文将 Ingham 关于 Riemann zeta 函数零点密度的经典估计改进为显式形式，并给出临界线四次矩带渐近正确主项的显式估计。
Estimates for the number of zeros of shifted combinations of completed Dirichlet series
本文为完备 Dirichlet 级数平移组合的一个子类在临界线上的零点数量提供了下界估计。
We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The result mirrors central limit theorems in random matrix theory that have been proved by Szegő, Spohn, and Soshnikov among others, and therefore provides support for the view that the zeros of the zeta function are distributed like the eigenvalues of a random matrix. A key ingredient in our proof is a simple bootstrapping of classical zero density estimates o…
We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally we apply these results to prove a gap theorem for power series which are remainders of the Grauert-Hironaka-Galligo Division Theorem.
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