Formulas for odd zeta values and powers of $π$
Marc Chamberland, Patrick Lopatto
本文找到了适用于所有非负整数$n$的$π^{2n+1}$和$ζ(2n+1)$快速收敛级数公式的一般规律，并给出了证明。
Plouffe猜想了对小$n$值的$π^{2n+1}$和$ζ(2n+1)$的快速收敛级数公式。我们找到了所有非负整数$n$值的一般规律并给出了证明。
Plouffe conjectured rapidly converging series formulas for $π^{2n+1}$ and $ζ(2n+1)$ for small values of $n$. We find the general pattern for all nonnegative integer values of $n$ and offer a proof.
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Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing the integral kernel associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.
Using a new construction of rational linear forms in odd zeta values and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values $ζ$(5), $ζ$(7),. .. , $ζ$(69).
For odd $N\geq 5$, we establish a short exact sequence about motivic double zeta values $ζ^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. From this we classify all the relations among depth-graded motivic double zeta values $ζ^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns relations among multiple zeta values of odd weight.
In this article, we obtain a transformation formula for the higher power of odd zeta values, which generalizes Ramanujan's formula for odd zeta values. We have also investigated many important applications, which in turn provide generalizations of the transformation formula of the Eisenstein series, Dedekind eta function etc.
We prove that the meromorphic continuations of the Ruelle and Selberg zeta functions considered by Bunke and Olbrich are of finite order not larger than the dimension of the underlaying compact, odd-dimensional, locally symmetric space.
The main goal of this article is to present an elementary proof of Ramanujan's identity for odd zeta values. Our proof solely relies on a Mittag-Leffler type expansion for hyperbolic cotangent function and Euler's identity for even zeta values.
In this paper, we give a proof of a conjecture made by Zagier about the inverse of some matrix related to double zeta values of parity $(\mathrm{even},\mathrm{odd})$. As a result, we obtain a family of Bernoulli number identities. We further generalize this family to a more general setting involving binomial coefficients of negative arguments.
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.
The generalized Bernoulli numbers and its relation with the Riemann zeta function at odd-integer arguments
本研究借助广义伯努利数推导正奇数自变量处黎曼ζ函数的新积分表示，进而得到相关有理数域上向量空间维数的判定准则。
For odd $k$, we give a formula for the relations between double zeta values $ζ(r,k-r)$ with $r$ even. This formula provides a connection with the space of cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. This is the odd weight analogue of a result by Gangl, Kaneko and Zagier. We also provide an answer of a question asked by Zagier about the left kernel of some matrix. Although the restricted sum statement fails in the odd weight case, we provide an asymptotical statement that replaces it. Our statement works more generally for restricted sums with any congruence condition on the first entry of the d…
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。
Recent developments pertaining to Ramanujan's formula for odd zeta values
本文综述了多位数学家对拉马努金奇数zeta值公式的贡献，并补充了Berndt和Straub综述中未涵盖的近期进展。
In this paper, we give elementary proofs of Zagier's formula for multiple zeta values involving Hoffman element and its odd variant due to Murakami. Zagier's formula was a key ingredient in the proof of Hoffman's conjecture. Moreover, using the same approach, we prove Murakami's formula for multiple $t$-values. This formula is essential in proving a Brown type result which asserts that each multiple zeta value is a $\mathbb{Q}$-linear combination of multiple $t$-values of the same weight involving $2$'s and $3$'s.
In this small note, we provide an elementary proof of the fact that infinitely many odd zeta values are irrational. For the first time, this celebrated theorem been proven by Rivoal and Ball--Rivoal. The original proof uses highly non-elementary methods like the saddle-point method and Nesterenko's linear independence criterion. Recently, Zudilin has re-proven a slightly weaker form of his important result that at least one of the odd zeta values $ζ(5),ζ(7),ζ(9)$ and $ζ(11)$ is irrational, by elementary means. His new main ingredient are certain 'twists by half' of hypergeometric series. Gener…
We show that there is a relationship between modular forms and totally odd multiple zeta values, by relating the matrix $E_{N,r}$, whose entries are given by the polynomial representations of the Ihara action, with even period polynomials. We also consider the matrix $C_{N,r}$ defined by Brown and give a new upper bound of the rank of $C_{N,4}$. This result gives support to the uneven part of the motivic Broadhurst-Kreimer conjecture of depth 4.
In this article, we introduce a recurrence formula which only involves two adjacent values of the Riemann zeta function at integer arguments. Based on the formula, an algorithm to evaluate $ζ$-values(i.e. the values of Riemann zeta function) at odd-integers from the two nearest $ζ$-values at even-integers is posed and proved. The behavior of the error bound is $O(10^{-n})$ approximately where $n$ is the argument. Our method is especially powerful for the calculation of Riemann zeta function at large argument, while for smaller ones it can also reach spectacular accuracies such as more than ten…
New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight $k$. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler integrals of holomorphic Eisenstein series.
Integral representations of the Riemann zeta function of odd argument
本研究借助Mikolas的双伽马函数表达式，推导出任意正整数p对应的奇数宗量黎曼ζ函数ζ(2p+1)的积分表示，给出了前几个宗量值的算例，部分结果与已有方法推导的结论一致。
A closed-form expression for zeta(2n+1) reveals a self-recursive function
欧拉发现了一个公式来表达所有偶数正整数参数的黎曼ζ函数的值。基于Dirichlet β函数的值、欧拉数和π，给出了所有奇整数参数的黎曼ζ函数的封闭表达式，揭示了奇数下黎曼ζ函数自递归性质的新证据.我们首次证明了黎曼ζ函数在奇数总是产生一个递归关系，是自递归的。
It is commonly known that $ζ(2k) = q_{k}\frac{ζ(2k + 2)}{π^2}$ with known rational numbers $q_{k}$. In this work we construct recurrence relations of the form $\sum_{k = 1}^{\infty}r_{k}\frac{ζ(2k + 1)}{π^{2k}} = 0$ and show that series representations for the coefficients $r_{k} \in \mathbb{R}$ can be computed explicitly.
We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $μ_2$. We find a basis for the depth-graded motivic double zeta values relative to $μ_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to $μ_2$ of odd weight. As an application of our main results, we prove Kaneko and Tasaka's conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka's conjecture in depth three. At last we formulate a conjecture which is related to sum odd m…
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.
It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values: $ζ(3),ζ(5),ζ(7),\cdots,ζ(s)$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ therein. The main new ingredient is an optimal design for the zeros of the auxiliary rational functions, which relates to the inverse …
We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.
La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs
我们提供了一个下界的维数的向量空间由1和黎曼ζ函数的值在第一个奇数。因此，Zeta函数在奇数处取无穷多个无理值。
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。
Totally odd depth-graded multiple zeta values and period polynomials
本文研究完全奇动机深度分级多重zeta值的关系，确定Brown矩阵的秩，给出$C_{N,3}$和$C_{N,4}$秩上界的新证明，并提出递归方法将一般问题归结为同构猜想。
We show for even positive integers $n$ that the quotient of the Riemann zeta values $ζ(n+1)$ and $ζ(n)$ satisfies the equation $$\frac{ζ(n+1)}{ζ(n)} = (1-\frac{1}{n}) (1-\frac{1}{2^{n+1}-1}) \frac{\mathcal{L}^\star(\mathfrak{p}_n)}{\mathfrak{p}_n'(0)},$$ where $\mathfrak{p}_n \in \mathbb{Z}[x]$ is a certain monic polynomial of degree $n$ and $\mathcal{L}^\star: \mathbb{C}[x] \to \mathbb{C}$ is a linear functional, which is connected with a special Dirichlet series. There exists the decomposition $\mathfrak{p}_n(x) = x(x+1) \mathfrak{q}_n(x)$. If $n = p+1$ where $p$ is an odd prime, then $\math…
We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $ζ(n)$. They naturally include the Beukers-Rhin-Viola integrals for $ζ(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objec…
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。
Linear independence of odd zeta values using Siegel's lemma
本文利用西格尔引理证明奇数ζ值的线性独立性，并改进了Rivoal在2001年的下界结果。
For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $ζ(i_1,...,i_k)$ with odd denominators. Like the multiple zeta values, the multiple $t$-values can be multiplied according to the rules of the harmonic algebra. Using this fact, we obtain explicit formulas for multiple $t$-values of repeated arguments analogous to those known for multiple zeta values. Multiple $t$-values can be written as rational linear combinations of the alternating or "colored" multiple zeta…
A note on the number of irrational odd zeta values, II
本研究通过结合Fischler-Sprang-Zudilin消元技术与Zudilin的Φₙ因子方法，改进了奇zeta值中无理数数量的下界，将此前结果以常数因子提升至至少1.284·√(s/log s)个。
On Witten multiple zeta-functions associated with semisimple Lie algebras V
本文研究了$G_2$型根系在正整数点的ζ函数值。在我们以前的工作中，我们考虑的情况下，所有的整数是偶数，但在本文中，我们证明了几个定理，其中包括的情况下，一些整数是奇数。本文还讨论了为什么我们可以处理包括奇数的情况的根本原因。
In this paper, we define and study a variant of multiple zeta values of level 2 (which is called multiple mixed values or multiple $M$-values, MMVs for short), which forms a subspace of the space of alternating multiple zeta values. This variant includes both Hoffman's multiple $t$-values and Kaneko-Tsumura's multiple $T$-values as special cases. We set up the algebra framework for the double shuffle relations (DBSFs) of the MMVs, and exhibits nice properties such as duality, integral shuffle relation, series stuffle relation, etc., similar to ordinary multiple zeta values. Moreover, we study …
In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. And from these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and three. In higher depth we conjecture that there are exact sequences of the same type. And we will show from three conjectures about depth-graded motivic Lie algebra we can nearly deduce the exact sequences conjectures in higher depth. At last we give a new proof of the result that the modulo zeta(2)$ version motivic double zeta values is generated by the to…
Arithmetic of linear forms involving odd zeta values
给出了（奇）zeta值下线性型的一般超几何构造。该构造允许恢复Rhin和Viola关于$ζ(2)$和$ζ(3)$的非理性度量的记录，以及解释Rivoal的“无穷多”结果（math.NT/0008051），并证明四个数$ζ(5)$、$ζ(7)$、$ζ(9)$和$ζ(11)$中至少有一个是非理性的。
Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values
该研究探讨Rogers-Ramanujan连分数的尖点展开对其普通与对数Mellin矩的控制作用，证明相关系数的代数性质，确定归一化矩生成函数的极点，并建立反射恒等式等结果。
The values of the Riemann zeta function at odd positive integers, $ζ(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$. Integral representations for $ζ(2n+1)$ are then deduced from the finite-part integral representation. Certain relations between $ζ(2n+1)$ and $ζ'(2n+1)$ are likewise deduced, from which integral representations for $ζ'(2n+1)$ are obtained.
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 note on odd zeta values over any number field and Extended Eisenstein series
本文推广了 Ramanujan 关于 Riemann zeta 函数的恒等式，建立任意数域上奇数 zeta 值的变换公式，并由此构造满足模变换的扩展 Eisenstein 级数，为研究奇数 zeta 值与 Lambert 级数提供新工具。
Evaluation of Riemann Zeta function on the Line $\Re(s) = 1$ and Odd Arguments
本文针对黎曼ζ函数在奇数参数和直线$\Re(s) = 1$上的评估，提出了一个具有指数收敛性的简单近似公式，并与文献中的其他公式进行了比较。
Distribution of irrational zeta values
本文改进了Ball-Rivoal定理，证明了对任何一个关于$ε>0$足够大的奇整数$a$，在3和$a$之间存在$[ \frac{(1-ε)\log a}{1+\log 2}]$奇数$s$，它们之间的距离至少为$a^ε$，在此点上黎曼ζ函数取$\Q$ -线性无关值.因此，如果整数$s$非常少，使得$ζ(s)$是无理数，那么它们是相当均匀分布的。证明涉及一系列的超几何型估计的鞍点方法，和推广到向量的Nesterenko的线性独立性准则。
Multiple zeta values, Padé approximation and Vasilyev's conjecture
索罗金在1996年给出了一个新的证明π是超越的。它是基于一个同时Padé近似问题，涉及某些多个多项式，在点1处的评估是多个zeta值等于pi的幂。在本文中，我们构造了一个相同的味道Padé逼近问题，并证明了它有一个唯一的解决方案的比例。在点1处，这提供了1和多个zeta值在扩展意义上的有理线性组合，其结果是奇数处的黎曼ζ函数的值。作为一个应用，我们得到了一个新的证明瓦西里耶夫猜想的任何奇数重量，关于某些超几何多重积分的显式评估，它首先证明了Zudilin在2003年。
The fact that the double zeta values at n and m can be written as a sum of products of two zeta values and of zeta value at m+n, whenever n+m is odd is due to Euler. We shall show a weak version of this result for the Galois l-adic realization.
We prove new relations on zeta function at even arguments and Dirichlet $L$ function at odd. The key idea is to make use of the Taylor series and partial fraction decomposition of cotangent and secant functions as we discuss in calculus and complex analysis.
In this paper, we investigate linear relations among regularized motivic iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ of depth two, which we call regularized motivic double zeta values. Some mysterious connections between motivic multiple zeta values and modular forms are known, e.g. Gangl--Kaneko--Zagier relation for the totally odd double zeta values and Ihara--Takao relation for the depth graded motivic Lie algebra. In this paper, we investigate so-called non-admissible cases and give many new Gangl--Kaneko--Zagier type and Ihara--Takao type relations for regularized motivi…
Zeta function of the projective curve $\pmb{aY^{2 l} = bX^{2 l} + cZ^{2 l}}$ over a class of finite fields, for odd primes $\pmb{l}$
设$p$和$l$是有理素数，使得$l$是奇数，并且$p$模$l$的阶是偶数。对于这样的素数$p$和$l$，以及对于$e=l, 2l$，我们考虑定义在有限域$\mathbf{F}_q$上的非奇异投影曲线$aY^e = bX^e + cZ^e$（$abc \neq 0$），使得$q=p^α\equiv 1(\bmod {e})$ .我们看到，费马曲线精确地对应于每一类（对于$e=l,2l$）中在$\mathbf{F}_q$上最大或最小的那些曲线。我们观察到，每个费马素数在特征为2的有限域上产生显式的极大和极小曲线。对于$e=2l$，我们明确地确定这类曲线的$ζ$ -函数，在$\mathbf{F}_q$上，作为变量$t$中的有理函数，对于不同的$a,b$和$c$的情况，在$\mathbf{F}_q^*$中。在每种情况下的$ζ$ -函数被认为满足这类具体曲线的韦尔定理（现在的定理）。
AGZT-Lectures on formal multiple zeta values
本研究利用后李结构与双重混洗李代数的自由奇生成猜想，推导形式多重zeta值的Hoffman基猜想，沿用Brown定理思路实现Zagier公式的形式语境提升。
We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values. This variant, which is regarded as the `shuffle counterpart' of Hoffman's `odd variant', exhibits nice properties such as duality, shuffle product, parity results, etc., like ordinary multiple zeta values. We also give some conjectures on relations between our values, Hoffman's values, and multiple zeta values.
Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any positive real number and $s$ is large enough in terms of $\varepsilon$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $\frac{1-\varepsilon}{1+\log2}\log s$ that follows from the Ball--Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.
An unpublished identity of Gosper restates a hypergeometric identity for odd zeta values in terms of an infinite product of matrices. We show that this correspondence runs much deeper, and show that many examples of WZ-accelerated series for zeta values lift to infinite matrix products. We also introduce a new matrix subgroup, the Gosper group, which all of our matrix products fall into.
Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed - II
对于Hurwitz zeta函数，得到了其高阶导数（相对于第一变量）在非正整数变元下的第二变量幂级数表达式，从而得到了Riemann zeta函数在正奇数变元下的快速递减级数表达式.进一步，我们得到了Dirichlet L-级数的相应结果.我们还统一证明了各种涉及Riemann zeta值的经典恒等式.
Fixing $t \in \mathbb{R}$ and a finite field $\mathbb{F}_q$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over $\mathbb{F}_q$ whose zeta function vanishes at $\frac{1}{2} + it$. Our upper bound is independent of $g$ and tends to $0$ as $q$ grows.
Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.
The cyclic sum formulas for multiple zeta and zeta-star values were respectively proved by Hoffman and Ohno, and Ohno and Wakabayashi. Kawasaki and Oyama obtained an analogous formulas for finite multiple zeta and zeta-star values. In this paper, we give a generalization of Kawasaki and Oyama's results.
Riemann Zeta Function. An Attempt to fathom Zeta(3)
已经在1734年欧拉发现了一个简短的明确公式的价值黎曼ζ函数Zeta（s）时，参数s等于一个正整数2n其中n= 1，2，3，。对于Zeta的奇正整数参数不存在这样的公式。本文特别讨论了Zeta（3）的情形。得到了Zeta（3）的一个公式，该公式除了一些已知的常数外，还包括一个快速收敛的无穷级数，其中每项都包含有理数和Pi的一个偶次幂。试图将这一系列转换成有限数量的条款，其中包含众所周知的常数是符合只有部分成功。给出了zeta（2n+1）的一般情况。
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.
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.
An algebraic proof of cyclic sum formula for multiple zeta values
我们介绍了一个代数制定的循环和公式的多个zeta值和多个zeta星值。我们还提出了一个代数证明的循环和公式的多个zeta值和多zeta星值，通过减少他们的川岛关系。
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