Arul Shankar, Jacob Tsimerman
We show how the Selberg $Λ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ to $n! = P(x)$ which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is $o(N)$. The proof uses techniques of Diophantine and Padé approximation.
Power Savings for Counting (Twisted) Abelian Extensions of Number Fields
本文研究了数域阿贝尔扩张的计数问题，证明了误差项具有显著的幂次节省，并在假设广义Lindelöf假设下达到平方根幂次节省，同时揭示了部分未知的低阶项结构。
A power-saving error term in counting $C_2 \wr H$ extensions of an arbitrary base field parametrized by discriminants
本文提出了一种方法，用于计算任意基域由判别式参数化的 $C_2 \wr H$ 扩张的显式主项和节能误差项，并放宽了对置换群 $H$ 的假设。
Counting $C_2 \wr S_4$ fields with a power saving error term
本文改进了Klüners关于 $C_2 \wr S_4$ 字段计数的结果，给出了更精确的幂节省误差项，并为其他群提供了新的界。
We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real multi-quadratic number fields.
In this note, we prove a power-saving remainder term for the function counting $S_3$-sextic number fields. We also give a prediction on the second main term. We also present numerical data on counting functions for $S_3$-sextic number fields. Our data indicates that our prediction is likely to be correct, and it also suggests the existence of additional lower order terms which we have not yet been able to explain.
We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Klüners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $…
We estimate the growth rate of the function which counts the number of torsion points of order at most $T$ on an algebraic subvariety of the algebraic torus $\mathbb G_m^n$ over some algebraically closed field. We prove a general upper bound which is sharp, and characterize the subvarieties for which the growth rate is maximal. For all other subvarieties there is a better bound which is power saving compared to the general one. Our result includes asymptotic formulas in characteristic zero where we use Laurent's Theorem, the Manin-Mumford Conjecture. However, we also obtain new upper bounds fo…
Counting elliptic curves with a cyclic $m$-isogeny over $\mathbb{Q}$
本文利用解析数论和Tauberian定理，给出了有理数域上具有循环$m$-同源的椭圆曲线在二次扭转和同构意义下的计数渐近式，并获得了幂次节省的误差项。
Secondary terms in the counting functions of quartic fields II
本文确定了满足局部条件的判别式有界 $S_4$-四次域的平滑计数，表示为两个主项之和并带有幂次节省误差项，并证明了四次环的类似结果及 Shintani zeta 函数的极点性质。
We improve the error terms of some estimates related to counting lattices from recent work of L. Fukshansky, P. Guerzhoy and F. Luca (2017). This improvement is based on some analytic techniques, in particular on bounds of exponential sums coupled with the use of Vaaler polynomials.
A link between error terms when counting smooth and rough numbers
研究光滑与粗糙数计数函数误差项之间的关系，并利用该关系得到de Bruijn近似Λ在光滑数计数中的误差上界。
Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.
Counting number fields using multiple Dirichlet series
本文利用多重狄利克雷级数的解析技术，提出了一种按任意惯性不变量排序的固定伽罗瓦群数域的计数方法，对无穷多新群证明了无条件结果，并在子凸性假设下证明了渐近增长率的存在性。
This paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-functions in rectangles.
An improved error term for counting $D_4$-quartic fields
本文改进了Cohen等人关于D₄-四次域计数公式的误差项，将其从原有结果提升至O(X^{5/8+ε})，并将类似结论推广到任意基域上的四次二面体扩张。
The error term in counting prime pairs
本研究建立了Hardy-Littlewood素数对计数猜想公式误差项与von Mangoldt函数构造的素数指数和$L^1$范数的关联，为数论中素数对分布的误差分析提供了新的联系视角。
Some examples of well-behaved Beurling number systems
该研究聚焦Beurling数系的幂次误差节省问题，证明了多类参数范围内$[α,β]$-系的存在性，并在黎曼猜想假设下构造出β<1/2的数系族，推进了Beurling广义素数理论的边界研究。
Pointwise improvement of the error term in the higher rank hyperbolic lattice counting problem
该研究针对高阶双曲格点计数问题，在Blomer等人成果基础上，证明了SL(n,ℤ)对应误差项的改进上界，还得到符合Rudnick–Phillips风格的Ω-结果。
We study fluctuations of the error term for the number of integer lattice points lying inside a 3-dimensional Cygan-Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.
We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}τ(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and …
We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Kor$\acute{a}$nyi Heisenberg-norm $\mathcal{N}_{4,1}(z,w)=(|z|^{4}+w^{2})^{1/4}$. In this case, our main theorem establishes the estimate $\mathcal{E}(x)=Ω_{\pm}(x^{\frac{1}{2}})$, where $\mathcal{E}(x)=\mathcal{S}(x)-\frac{π^{2}}{2}x$ is the error term arising in the lattice point counting problem, $\mathcal{S}(x)$ is given by $$\mathcal{S}(x)=\sum_{0\leq m^2+n^2<\, x}r_2(m)$$ and $r_2(m)=|\{a,b\in\mathb…
Counting square-free values of random polynomials
本文研究了随机多项式平方自由值的计数问题，证明了平均误差项与主项的四次方根关系。
Secondary terms in the first moment of $|{\rm Sel}_2(E)|$
针对高度有界于X的椭圆曲线E的2-Selmer群$|{\rm Sel}_2(E)|$计数函数，首次证明存在$X^{3/4}$阶次要项及幂节省误差项，改进了Bhargava--Shankar的$o(X^{5/6})$误差项结果。
Uniform bounds for lattice point counting and partial sums of zeta functions
本文证明格点球内计数渐近式与 Landau 型 zeta 函数部分和渐近式的一致版本，使误差项隐含常数分别只依赖维数和函数方程形状。
Let $\mathbf{K}$ be a number field and $\mathfrak{q}$ an integral ideal in $\mathcal{O}_{\mathbf{K}}$. A result of Tatuzawa from 1973, computes the asymptotic (with an error term) for the number of ideals with norm at most $x$ in a class of the narrow ray class group of $\mathbf{K}$ modulo $\mathfrak{q}$. This result bounds the error term with a constant whose dependence on $\mathfrak{q}$ is explicit but dependence on $\mathbf{K}$ is not explicit. The aim of this paper is to prove this asymptotic with a fully explicit bound for the error term.
An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields
本文提出仅依赖 L-函数平均大小的显式陶伯定理，并将其应用于证明若干循环扩张数域计数问题中的平方根节省误差界。
The hyperbolic lattice counting problem in large dimensions
本文研究大维度双曲空间中余紧格作用下的圆问题误差项平均行为，给出局部平均上界并证明均值与二阶矩的Ω结果。
We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly t…
Correlations of error terms for weighted prime counting functions
本文研究加权素数计数函数归一化误差项之间的相关性，证明某些持续不等式等价于黎曼假设，并在RH与LI假设下计算了误差项符号关系的对数密度。
We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
Square root cancellation in the hyperbolic lattice counting problem over cocompact groups
本文针对余紧Fuchsian群改进了双曲格点计数误差项的Selberg界，达到几乎处处最优的平方根上界，并推广至高维，同时改进二阶矩界并否证Phillips-Rudnick猜想。
We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these approximations are at least as good as the logarithmic integral approximation. As a corollary, we provide some reformulations of the Riemann hypothesis in terms of the prime counting function and the harmonic numbers.
The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_k(n)^2$, where $P_k(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_k(n)^2 e^{-n/X}$ and the Laplace transform $\int_0^\infty P_k(t)^2 e^{-t/X}dt$, in dimensions $k \geq 3$. We also obtain main terms and power-saving error terms for the sharp sums $\…
An $Ω$-Result for the Counting of Geodesic Segments in the Hyperbolic Plane
本文针对余紧 Fuchs 群中固定闭测地线像的计数问题，证明了误差项的下界为 $\Omega_\delta(X^{1/2}(\log\log X)^{1/4-\delta})$，其中 $X=\cosh R$。
On the counting function of cubic function fields
本文研究三次函数域的计数函数，推导出包含次级项和最优误差项的渐近公式，并利用数的几何方法得到轨道数精确公式，通过Artin L-函数密度证明精细计数误差项的无条件下界。
An incremental approximation to the prime-counting function via polygonal numbers
提出一种基于广义三角数结构化非均匀划分的素数计数函数π(x)近似算法，具有O(1)均摊更新复杂度和O(√x)总复杂度，通过数值实验修正项达到与经典解析近似相当的精度。
On the counting function of square-full numbers
在黎曼假设及ζ(s)零点为单零点的前提下，研究全平方数的分布，改进了其计数函数的误差项并补充了额外主项。
We prove an asymptotic formula with power saving error term for a certain triple divisor sum.
Counting $2\times2$ integer matrices with fixed trace and determinant
该研究利用二次根的Weyl和的和的界，对带固定特征多项式的2×2整数矩阵数目做带光滑权的计数，给出主项与误差项。
The hyperbolic lattice point problem asks to estimate the size of the orbit $Γz$ inside a hyperbolic disk of radius $\cosh^{-1}(X/2)$ for $Γ$ a discrete subgroup of $\hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for $\hbox{PSL}_2(Z)$. The result is that the error term can be improved to $O(X^{7/12+ε})$. The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maaß cusp forms and estimat…
Explicit counting of ideals in number fields of arbitrary degree
本文利用数论几何方法，研究数域中积分理想数量的显式估计，通过最小化次数对误差项的影响，避免高阶项，并利用刘维尔空间中的基本域理论和施密特的分拆技巧改进误差估计。
Counting $2\times 2$ matrices with fixed determinant and bounded coefficients
本文改进了Afifurrahman关于固定行列式n和系数有界矩阵计数的结果，给出了更优的误差项，并证明了1/2+δ指数无法降低。
Counting cusp forms by analytic conductor
该研究针对数域上${\rm GL}_n$的酉尖点自守表示通用族，证明了解析导子截断下的族大小渐近公式，给出首项常数几何解释与Sato–Tate测度猜想，还得到一致Weyl定律与非tempered离散谱定量界。
We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box, which generalises a counting theorem of Skriganov. The error term is expressed in terms of a certain function $ν(Γ^\perp,\cdot)$ of the dual lattice $Γ^\perp$, and we carefully analyse the relation of this quantity with $ν(Γ,\cdot)$. In particular, we show that $ν(Γ^\perp,\cdot)=ν(Γ,\cdot)$ for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. Finally, we …
Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^2 \times \mathbb{P}^2$ with bihomogeneous coordinates $[x_1:x_2:x_3],[y_1:y_2,y_3]$ and in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with multihomogeneous coordinates $[x_1:y_1],[x_2:y_2],[x_3:y_3]$ defined by the same equation $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$. We thus improve on recent work of Blomer, Brüdern and Salberger \cite{BBS} and provide a different proof based on…
Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^α)$, for an explicit $α<1$.
We use a dictionary between lattice point counting inside dilated d-dimensional ellipsoids (Euclidean counting) and counting of lifts of a closed horosphere that intersect a ball of increasing radius, to obtain two types of results. Firstly, via an $L^2$-integral error estimate for Euclidean counting, we prove effective equidistribution results for a family of expanding horospheres in the locally symmetric space SO(d) \ SL(d,R) / SL(d,Z). Secondly, we derive from uniform error estimates in Euclidean counting, error terms for counting SL(d,Z)-orbit points in a certain increasing family of subse…
We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{α,A}((z,t)) = \left(|z|^α+ A |t|^{α/2}\right)^{1/α}$, for $α\ge 2$ and $A>0$. This natural family includes the canonical Cygan-Korányi norm, corresponding to $α=4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $α=2$ is the best possible, in all dimensions.
Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control
本文研究了有限域上涉及线性无关多项式的多维多项式递进，证明了幂次节省界限，并通过定量 PET 归纳、Gowers 盒范数拼接、平滑技术和多维降次论证，为两两独立多项式实现了 Gowers 范数控制。
A Rigorous Error Bound for the TG Kernel in Prime Counting
本文在显式公式框架内利用截断高斯核为素数计数建立了严格误差界，证明在足够大的自变量下误差全局低于1/2，仅需约1200个非平凡zeta零点即可实现秒级精确计算π(x)，为计算数论提供了实用新算法。
A strong form of the Manin-Peyre conjecture with a power saving error term is proved for a certain cubic fourfold.
We study the error of the number of unimodular lattice points that fall into a dilated and translated parallelogram. By using an article from Skriganov, we see that this error can be compared to an ergodic sum that involves the discrete geodesic flow over the space of unimodular lattices. With the right normalization, we show, by using tools from a previous work of Fayad and Dolgopyat, that a certain point process converges in law towards a Poisson process and deduce that the ergodic sum converges towards a Cauchy centered law when the unimodular lattice is distributed according to the normali…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。