Wadim Zudilin
We present several results on the number of irrational and linear independent values among $ζ(s),ζ(s+2),...,ζ(s+2n)$, where $s>2$ is an odd integer and $n>0$ is an integer. The main tool in our proofs is a certain generalization of Rivoal's construction (math.NT/0008051, math.NT/0104221).
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
The proof of the irrationality of Zeta(5) is a long standing open problem, but here only the case of Zeta(4) = (Pi^4)/90 is considered. The present paper suggests an approach for the irrationality of Zeta(4) along the lines of those known for proving the irrationality of Zeta(3).
A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction yields Apéry's approximations to $ζ(2)$ and $ζ(3)$, and for larger $n$, an infinite family of small linear forms in multiple zeta values with an interesting algebraic structure. It also contains a generalisation of the linear forms used by Ball and Rivoal to prove that infinitely many odd zeta values are irrational.
The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), ζ_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of this result. In parallel, T. Rivoal has just obtained, in the complex case, some Pade approximants of Lerch functions. It is this work which, transposed to C_p, enables us to obtain results of irrationality and linear independence.
In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used …
We give a proof of the irrationality of the $p$-adic zeta-values $ζ_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show irrationality of some other $p$-adic $L$-series values, and values of the $p$-adic Hurwitz zeta-function.
On the irrationality of certain $p$-adic zeta values
本文建立了 Zudilin 定理的 $p$-进类似物，证明了对于任意素数 $p \geqslant 5$，在特定区间内存在奇数 $i$ 使得 $p$-进 zeta 值 $ζ_p(i)$ 为无理数。
A motivated q-extension of the values of the Riemann zeta function at positive integers is presented. Several irrationality and transcendence results as well as new general problems for these q-zeta values are stated.
A new upper bound for the irrationality exponent of zeta(3)
本文通过应用Rhin-Viola定理5.1于新的积分方向，将zeta(3)的无理性指数上界改进至5.5138800，优于此前保持25年的最佳结果，并指出Rhin-Viola点并非全局最优。
Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain functional equations arising from Eichler integrals are replaced by the notion of overconvergent p-adic modular forms, and the periods themselves arise not as coefficients of period polynomials but as constant terms of p-adic Eisenstein series. We prove that the analogue of zeta(3) is irrational for p = 2 and 3, as well as the 2-adic analogue of Catalan's constant.
On the irrationality of certain $2$-adic zeta values
本文证明了 Kubota-Leopoldt $2$-adic zeta 函数在特定区间内存在无理数值，通过构造显式有理函数并利用 Volkenborn 积分结合二项式系数操作实现。
There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $ζ(2)$ and $ζ(3)$, as well as to explain Rivoal's "infinitely-many" result (math.NT/0008051) and to prove that at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$, and $ζ(11)$ is irrational.
New formulas for approximation of zeta-constants were derived on the basis of a number-theoretic approach constructed for the irrationality proof of certain classical constants. Using these formulas it's possible to approximate certain zeta-constants and their combinations by rational fractions and construct a method for their evaluation.
Available proofs of result of the type 'at least one of the odd zeta values $ζ(5),ζ(7),\dots,ζ(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are however counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result - a proof whose technical ingredients are limited to the prime numbe…
A $q$-analogue of the Koecher-Leshchiner generating function of odd zeta values
本文推导了奇ζ值生成函数的q-类比，提供了Apéry证明ζ(3)无理数性所用加速级数的q-版本。
In a recent paper with Sprang and Zudilin, the following result was proved: if $a$ is large enough in terms of $\varepsilon>0$, then at least $2^{(1-\varepsilon)\frac{\log a}{\log \log a}}$ values of the Riemann zeta function at odd integers between $3$ and $a$ are irrational. This improves on the Ball-Rivoal theorem, that provides only $\frac{1-\varepsilon}{1+\log 2} \log a$ such irrational values -- but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both $L$-functions of Dirichlet characters and Hurwitz zeta…
This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Apéry's Theorem (1978) : $ζ(3)$ is irrational. The second section is devoted to a variant of the proof, published by Rivoal and Ball-Rivoal, that infinitely many $ζ(2n+1)$ are irrational. The end of this text deals with more quantitative statements.
We present a brief survey of the methods used in deducing upper estimates for irrationality measures of the logarithm values. We particularly expose the best known estimates for $\log2$ (due to E. Rukhadze), $π$ (due to M. Hata) and $\log3$ (due to G. Rhin).
We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.
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。
Apéry-Like Sums and Colored Multiple Zeta Values
本文综述类Apéry和（Apéry无理性证明中两个和的多变量推广）的近期进展，介绍其与有色多重Zeta值的关联及在多圈费曼图ε展开中的作用，总结计算方法并证明孙智伟的猜想恒等式。
We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm 1/2,\pm 1/3,\pm 1/4,...
The fourth moment of the Hurwitz zeta function
该研究针对平移参数为无理性指数小于3的无理数的Hurwitz zeta函数，给出其临界线上四阶矩的精确上界，确定了0≤k≤2时2k阶矩的量级，推测其临界线上值分布为高斯分布。
Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $ι_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding $ι_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \tfrac{1}{z}K\left[\left[\tfrac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Padé approximation, the special values $\mathcal{M}(g)(α)$ are convergent in $K_p$ and irrationa…
We present a hypergeometric construction of rational approximations to $ζ(2)$ and $ζ(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $ζ(2)$ and $ζ(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S. Fischler.
We prove that the formula giving the exact value of the irrationality exponent of regular continued fractions remains valid for semi-regular continued fractions satisfyiong certain conditions.
We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed.
By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by {logS(n)} = d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational number. A corollary is that if {logS(n)} > 1/2^n infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 0 < n < 2500 and n = 10000) and gamma …
We introduce a one-parameter family of series associated to the Riemann $ζ$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the parameter, where almost all is with respect to any sufficiently nice measure. We also give similar results for the Euler--Mascheroni constant, for $\sum_{n=1}^\infty \frac{1}{n^n}$ and for $\sum_{n=1}^\infty \frac{1}{n! +1}$. Finally, specialising the criteria used, we give some new criteria for the irrationality of $ζ(k)$, the Euler--Mascheroni constant and the latter t…
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。
On the Connection Between Irrationality Measures and Polynomial Continued Fractions
本文探讨了多项式连分数与无理数度量之间的联系，扩展了阿佩里工作，提出关于PCFs生成的DAs的一般结论和猜想，并应用于基本常数的逼近。
Let $F(x)$ be an analytical, real valued function defined on a compact domain $\mathcal {B}\subset\mathbb{R}$. We prove that the problem of establishing the irrationality of $F(x)$ evaluated at $x_0\in \mathcal{B}$ can be stated with respect to the convergence of the phase of a suitable integral $I(h)$, defined on an open, bounded domain, for $h$ that goes to infinity. This is derived as a consequence of a similar equivalence, that establishes the existence of isolated solutions of systems equations of analytical functions on compact real domains in $\mathbb{R}^p$, if and only if the phase of …
A unified proof of the irrationality of the special values L(n, X), n > 1 an integer, of the beta L-function is put forward in this note. The first case of n = 2 seems to confirm that the Catalan constant L(2, X) is an irrational number.
Many $p$-adic odd zeta values are irrational
本文证明充分大的奇整数范围内有相当数量的 $p$ 进奇数 zeta 值无理，并借助新无理性判据解决线性形式非零性难题。
This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the general…
We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided.
For different values of $γ\geq 0$, analysis of the end behavior of the sequence $a_n = \cos (n)^{n^γ}$ yields a strong connection to the irrationality measure of $π$. We show that if $\limsup |\cos n|^{n^2} \neq 1$, then the irrationality measure of $π$ is exactly 2. We also give some numerical evidence to support the conjecture that $μ(π)=2$, based on the appearance of some startling subsequences of $\cos(n)^n$.
In this paper, improving the method of Allouche \emph{et al.} \cite{APWW98}, we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence module 2 is periodic with period 10 which answers Coon's conjecture \cite{CV12}. Then we extend Bugeaud's method \cite{Bugeaud11} to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2.
Special values of spectral zeta functions of one-, two-photon quantum Rabi models and non-commutative harmonic oscillators
本文研究了量子 Rabi 模型和非交换谐振子的谱 zeta 函数在正整数处的显式表达式，并探讨了其与 Beukers 积分的联系。
We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{α_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vertα_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $α= [0;α_1, α_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.
We have calculated numerically geometrical means of the denominators of the continued fraction approximations to the Brun constant B2. We get values close to the Khinchin constant. Next we calculated the n-th square roots of the denominators of the n-th convergents of these continued fractions obtaining values close to the Khinchin-Levy constant. These two results suggests that B2 is irrational, supporting the common believe that there is an infinity of twins.
Diophantine approximation of multiple zeta-star values
本文针对多重zeta星值集合建立了丢番图逼近的经典结果，包括判定准则、零一律及一个加强的猜想。
In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even a…
We give an explicit representation for the sums of multiple zeta-star values of fixed weight and height in terms of Riemann zeta values.
Permutation of values of irrationality measure functions
研究了n个无理数不等式测度函数的排列性质，证明在极端情况下，排列结果形成π的轨道。
The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values for all non-zero algebraic values of $z$ ($z \ne 1$ in the case of $\ln{z}$). On the other hand, not even an irrationality proof is known for some numbers like $\,e^e$, $\,π^e$, $\,π^π$, $\,\lnπ$, $\,π+ e\,$ and $\,π\, e$, though it is well-known that at least one of the last two numbers is irrational. In this note, I first derive a more genera…
The Tijdeman-Zagier conjecture states no integer solution exists for $A^X+B^Y=C^Z$ with positive integer bases and integer exponents greater than 2 unless gcd$(A,B,C)>1$. Any set of values that satisfy the conjecture correspond to a lattice point on a Cartesian graph which subtends a line in multi-dimensional space with the origin. Properties of the slopes of these lines in each plane are established as a function of coprimality of terms, such as irrationality, which enable us to explicitly prove the conjecture by contradiction.
Apéry-type approximations and irrationality measures for certain $q$-series
该研究构造三参数有理逼近，证明拉马努金θ函数等三类q级数在特定点的值为无理数，给出其无理性测度上界，并发现其中一个逼近与Coussement–Smet的帕德逼近一致。
Irrationality of finite logarithms in a congruence-class adèle ring
该文在$abc$猜想成立前提下，将有限对数无理性的已有结果扩展到素数受限为$p\equiv 1\bmod m$的算术级数情形，还证明其无法为二次无理数。
Perspectives on the arithmetic nature of the ratios $ζ(2n + 1)/π^{2n+1}$ and $β(2n)/π^{2n}$
本文通过分析框架探讨了ζ和β函数在整数处的值，并提出线性无关猜想以证明相关比值的无理数性。
We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.
We prove integral representations of the approximation forms in zeta values constructed in arXiv:1801.09895 and arXiv:1803.08905.
An explicit formula for the height-one multiple zeta values was proved by Kaneko and the second author. We give an alternative proof of this result and its generalization. We also prove its counterpart for the finite multiple zeta values.
We prove a new linear relation for multiple zeta values. This is a natural generalization of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.
The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.
We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height.
Cyclotomic symmetric multiple zeta values span the space of cyclotomic multiple zeta values
该研究证明了不同学者独立提出的分圆对称多重zeta值可张成模$\pi i$的分圆多重zeta值空间，为分圆多重zeta值的结构研究提供了关键结论。
In this paper we consider iterated integrals of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst \cite{BBBL1997}, and give some general results. Furthermore, we discuss Kaneko-Yamamoto multiple zeta values, and establish some relations between it and multiple zeta values. Finally, we establish a linear re…
Difference of irrationality measure functions
本文研究无理数测度函数的差值问题，在已有关于与τ等价数的最优结果基础上，证明了当至少一个无理数不与τ等价时，差值下界的新常数并验证其最优性。
The derivation relation is a well known relation among multiple zeta values, which was first obtained by Ihara, Kaneko and Zagier. The analogous formula for finite multiple zeta values, which we call the derivation relation for finite multiple zeta values, was conjectured by the third author and proved by the second author. In this paper, we show these two kinds of derivation relations are respectively equivalent to the Ohno type relations for multiple zeta values and finite multiple zeta values. We also reprove these derivation relations in several different ways.
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。