Extreme values for iterated integrals of the logarithm of the Riemann zeta-function

Duality/Sum formulas for iterated integrals and their application to multiple zeta values

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given.

Evaluation of iterated log-sine integrals in terms of multiple polylogarithms

It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple polylogarithms and multiple zeta values in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals.

On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: denseness

We consider iterated integrals of $\logζ(s)$ on certain vertical and horizontal lines. Here, the function $ζ(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log ζ(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of…

Iterated integrals on products of one variable multiple polylogarithms

In this paper we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals.

Iterated integrals and unipotent periods on families of marked elliptic curves

We describe iterated integrals as unipotent periods on families of marked elliptic curves in terms of multiple zeta values and elliptic multiple zeta values.

形式荒川-金子型zeta值经由迭代积分的统一方法

A Unified Approach to Formal Arakawa-Kaneko Type Zeta Values via Iterated Integrals

本文通过迭代积分方法引入形式多重对数函数和一般形式荒川-金子zeta值，建立对偶与求和公式，统一推导已知结果，展现统一性。

Multiple Zeta Values and Ideles

In this paper we give two idelic representations of the multiple zeta values - one using iterated integrals over the finite ideles and the other using iterated integrals over the idele class group. Each of the representations leads to a shuffle relation. Thus, we recover in a unified way the two types of shuffle relations of multiple zeta values via the iterated integrals over finite ideles and via iterated integrals over the idele class group.

Iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and a class of relations among multiple zeta values

In this paper we consider iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and define a class of $\mathbb{Q}$-linear relations among them, which arises from the differential structure of the iterated integrals with respect to $z$. We then define a new class of $\mathbb{Q}$-linear relations among the multiple zeta values by taking their limits of $z\rightarrow1$, which we call \emph{confluence relations} (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust …

Multiple zeta values and iterated Eisenstein integrals

Brown showed that the affine ring of the motivic path torsor $π_1^{\text{mot}}(\mathbb{P}^1 \backslash \left\{0,1,\infty\right\}, \vec{1}_0, -\vec{1}_1)$, whose periods are multiple zeta values, generates the Tannakian category $\mathsf{MT}(\mathbb{Z})$ of mixed Tate motives over $\mathbb{Z}$. Brown also introduced multiple modular values, which are periods of the relative completion of the fundamental group of the moduli stack $\mathcal{M}_{1,1}$ of elliptic curves. We prove that all motivic multiple zeta values may be expressed as $\mathbb{Q}[2 πi]$-linear combinations of motivic iterated Ei…

Multiple $T$-values and iterated log-tangent integrals

Multiple $T$-values, a variant of multiple zeta values of level two, were introduced and studied by Kaneko and Tsumura. This paper will introduce iterated log-tangent integrals and discuss their relations with multiple $T$-values. We will use ideas from the author's previous work on multiple zeta values and iterated log-sine integrals to do so.

正则化精炼对称多重 zeta 值的 Ohno 关系

Ohno relation for regularized refined symmetric multiple zeta values

本文将 Ohno 关系从精炼对称多重 zeta 值进一步推广到正则化精炼对称多重 zeta 值，即从 $0$ 到 $0$ 的非容许迭代积分情形。

On the law of the iterated logarithm for permuted lacunary sequences

It is known that for any smooth periodic function $f$ the sequence $(f(2^kx))_{k\ge 1}$ behaves like a sequence of i.i.d.\ random variables, for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting $(f(2^kx))_{k\ge 1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k)_{k\ge 1}$, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any p…

Iterated integrals, multiple zeta values and Selberg integrals

Classical multiple zeta values can be viewed as iterated integrals of the differentials $\frac{dt}{t}, \frac{dt}{1-t}$ from $0$ to $1$. In this paper, we reprove Brown's theorem: For $a_i, b_i, c_{ij}\in \mathbb{Z}$, the iterated integral of the form \[ \mathop{\int\cdots \int}\limits_{0<t_1<\cdots<t_N<1}\prod_i t_i^{a_i}(1-t_i)^{b_i} \prod_{i<j}(t_j-t_i)^{c_{ij}}dt_1\cdots dt_N \] is a $\mathbb{Q}$-linear combination of multiple zeta values of weight $\leq N$ if convergent. What is more, we show that if $p_i(t), 1\leq i\leq N, $ are in a $\mathbb{Q}\left[t,1/t, 1/(1-t)\right]$-algebra generat…

Elliptic Double Zeta Values

We study an elliptic analogue of multiple zeta values, the elliptic multiple zeta values of Enriquez, which are the coefficients of the elliptic KZB associator. Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. In this paper, we prove that the $\mathbb{Q}$-span of these elliptic multiple zeta values forms a $\mathbb{Q}$-algebra, which is naturally filtered by the length and is conjecturally graded by the w…

Law of the iterated logarithm for a random Dirichlet series

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(σ)=\sum_{n=1}^\infty X_nn^{-σ}$. We prove that the following holds almost surely \begin{equation*} \limsup_{σ\to 1/2^+}\frac{F(σ)}{\sqrt{2\mathbb E F(σ)^2\log\log \mathbb E F(σ)^2}}=1. \end{equation*}

The alternating block decomposition of iterated integrals, and cyclic insertion on multiple zeta values

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lisoněk states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ ζ(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ π$. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 ζ(3,3,\{2\}^m) - ζ(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see …

Selberg类中$L$-函数对数的迭代积分的普适性

Universality for the iterated integrals of logarithms of $L$-functions in the Selberg class

本文证明了在平行于实轴的某直线上，Selberg类中$L$-函数对数的迭代积分满足普适性定理。

Explicit Relations between Multiple Zeta Values and Related Variants

In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Apéry-type variant of MZSVs $ζ^\star_B({\bf k})$ (called multiple zeta $…

Iterated primitives of logarithmic powers

The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these polynomials is established.

Mean Values of the Logarithmic Derivative of the zeta Function and the GUE Hypothesis

The GUE Hypothesis, which concerns the distribution of zeros of the Riemann zeta-function, is used to evaluate some integrals involving the logarithmic derivative of the zeta-function. Some connections are shown between the GUE Hypothesis and other conjectures.

Zinbiel algebras and multiple zeta values

We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values.

来自 zeta 生成元的非全纯模形式

Non-holomorphic modular forms from zeta generators

本文利用 zeta 生成元分析等变迭代 Eisenstein 积分，从奇 Riemann zeta 值推导出多重 zeta 值与模图形式的出现，并构造三重 Eisenstein 积分的模补全，在 Brown 框架下显式实现模图形式。

Shuffle product of desingularized multiple zeta functions at integer points

In this paper, we investigate the ``shuffle-type'' formula for special values of desingularized multiple zeta functions at integer points. It is proved by giving an iterated integral/differential expression for the desingularized multiple zeta functions at integer points.

On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces

We derive approximate formulas for the logarithmic de- rivative of the Selberg and Ruelle zeta functions over compact, even- dimensional, locally symmetric spaces of rank one. The obtained for- mulas are given in terms of the zeta-singularities.

Exponential moments of the logarithm of the Riemann zeta-function twisted by arguments

We discuss moments of the Riemann zeta-function in this paper. The purpose of this paper is to give an upper bound of exponential moments of the logarithm of the Riemann zeta-function twisted by arguments. Our results contain an improvement of Najnudel result for exponential moments of the argument of the Riemann zeta-function and an unconditional upper bound of the moments.

On a generalization of Chen's iterated integrals

Chen's iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive and a (non-classical) multiplicative iterative property, in addition to a comultiplication formula. This theory is developed in the first part of the paper, after which various applications are discussed, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the exist…

On logarithmic derivatives of zeta functions in families of global fields

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer-Siegel theorem both for number fields and function fields.

A new proof of the duality of multiple zeta values and its generalizations

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for ($q$-)multiple zeta values.

Iterated integrals associated with colored rooted trees

In this paper, we introduce iterated integrals associated with colored rooted trees and give proofs for the shuffle relations for $\boldsymbol{p}$-adic finite and $t$-adic symmetric polylogarithms. This method generalizes the theory of the finite multiple zeta values associated with $2$-colored rooted trees introduced by Ono.

On the value-distributions of logarithmic derivatives of Dedekind zeta functions

We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any algebraic number field. We construct the density functions as the Fourier inverse transformations of certain functions represented by infinite products that come from the Euler products of the Dedekind zeta functions.

Relations between elliptic multiple zeta values and a special derivation algebra

We investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length. Our method is based on representing elliptic multiple zeta values as iterated integrals over Eisenstein series and exploiting the connection with a special derivation algebra. Its commutator relations give rise to constraints on the iterated integrals over Eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives. Conversely, the above connection suggests apparently new r…

Integral expressions for Schur multiple zeta values

Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.

Arborified multiple zeta values

We describe some particular finite sums of multiple zeta values which arise from J. Ecalle's "arborification", a process which can be described as a surjective Hopf algebra morphism from the Hopf algebra of decorated rooted forests onto a Hopf algebra of shuffles or quasi-shuffles. This formalism holds for both the iterated sum picture and the iterated integral picture. It involves a decoration of the forests by the positive integers in the first case, by only two colours in the second case.

Deformation of Multiple Zeta Values and Their Logarithmic Interpretation in Positive Characteristic

Pellarin introduced the deformation of multiple zeta values of Thakur as elements over Tate algebras. In this paper, we relate these values to a certain coordinate of a higher dimensional Drinfeld module over Tate algebras which we will introduce. Moreover, we define multiple polylogarithms in our setting and represent deformation of multiple zeta values as a linear combination of multiple polylogarithms.

The distribution of the logarithmic derivative of the Riemann zeta-function

We investigate the distribution of the logarithmic derivative of the Riemann zeta-function on the line Re(s)=σ, where σ, lies in a certain range near the critical line σ=1/2. For such σ, we show that the distribution of ζ'/ζ(s) converges to a two-dimensional Gaussian distribution in the complex plane. Upper bounds on the rate of convergence to the Gaussian distribution are also obtained.

关于zeta函数和$L$-函数的对数导数的Omega定理

Omega theorems for logarithmic derivatives of zeta and $L$-functions

本文为黎曼zeta函数和Dirichlet L-函数的对数导数建立了新的Omega定理，改进了Landau、Bohr-Landau及Lamzouri的先前结果。

On the behavior of the logarithm of the Riemann zeta-function

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{ζ(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the relation by some previous works. For example, Littlewood showed an upper bound of $\log{ζ(1/2 + it)}$ by assuming the Riemann Hypothesis in 1924. One of our results reveals that Littlewood's upper bound can be proved without assuming a hypothesis as strong as the Riemann Hypothesis.

Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values

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…

全实分圆域上的迭代余切差分积分

Iterated cotangent-difference integrals over totally real cyclotomic fields

本文研究全实分圆域上的迭代余切差分积分，将其表示为分圆多重zeta值，并证明这些积分是混合Tate动机的周期，生成Hopf子代数。

Identities for the multiple zeta (star) values

In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying these formulas, we can prove that multiple zeta star values whose indices are the sequences $(\bar 1,\{1\}_m,\bar 1)$ and $(2,\{1\}_m,\bar 1)$ can be expressed polynomially in terms of zeta values, polylogarithms and $\ln2$. Finally, we also evaluate several restricted sum formulas involving multiple zeta values.

Iterated integrals of modular forms and noncommutative modular symbols

The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper halfplane, eventually multiplied by $z^{s-1}$, along geodesics connecting two cusps. This setting generalizes simultaneously the theory of modular symbols and that of multiple zeta values

A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis II

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical resear…

Integrals of logarithmic functions and alternating multiple zeta values

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form \[ζ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},\bar 1,{\left\{ 1 \right\}_{k - 1}}} ),\ (k,m\in \mathbb{N})\] for $m=1$ or $k=1$, and \[ζ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},p,{\left\{ 1 \right\}_{k - 1}}}),\ (k,m\in\mathbb{N})\] for $p=1$ and $2$, satisfy certain recurrence relations which allow us to write them in terms of zeta values, pol…

Elliptic multiple zeta values and one-loop superstring amplitudes

We investigate iterated integrals on an elliptic curve, which are a natural genus-one generalization of multiple polylogarithms. These iterated integrals coincide with the multiple elliptic polylogarithms introduced by Brown and Levin when constrained to the real line. At unit argument they reduce to an elliptic analogue of multiple zeta values, whose network of relations we start to explore. A simple and natural application of this framework are one-loop scattering amplitudes in open superstring theory. In particular, elliptic multiple zeta values are a suitable language to express their low …

A symbolic approach to the poly-Bernoulli numbers

We present a symbolic representation for the poly-Bernoulli numbers. This allows us to prove several new iterated integral representations for the poly-Bernoulli numbers, including an integral transform of the Bernoulli-Barnes numbers. We also deduce some new recurrences for the poly-Bernoulli numbers. Finally, we use these results to present a new iterated integral representation for the Arakawa-Kaneko zeta function, including a nonlinear integral transform of the Barnes zeta function.

临界带内 Selberg zeta 函数对数导数的平方积分

Square integrals of the logarithmic derivatives of Selberg's zeta functions in the critical strip

本文在已有对数导数上界工作的基础上，研究 Selberg zeta 函数对数导数的平方积分在模群、余紧算术群及其子群情形下的增长性质。

A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis III. Exponential weight functions

This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type, which this time involve exponential functions, are established, and for the first time we have found equalities involving the integrals of the logarithm of the Riemann zeta-function taken exclusively along the real axis. Some of the equalities we hav…

Logarithmic Fourier integrals for the Riemann Zeta Function

We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line ${Re}(s)=\frac12$ and Blaschke zeta zeroes.

Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series

In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …

Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line

Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number $a>0$, \begin{align*} \lim_{a \to 0^+}\lim_{T \to \infty} \frac{(2a)^{2K-1}}{T (\log T)^{2K}} \int_{T}^{2T} \left|\frac{ζ'}ζ\left(\frac{1}{2}+\frac{a}{\log T}+it\right)\right|^{2K} dt = \binom{2K-2}{K-1}. \end{align*} When $K=1$, this was essentially a result of Goldston, Gonek and Montgomery.

关于zeta函数及其对数导数的联合二阶矩

On the joint second moment of zeta and its logarithmic derivative

本文在假设Riemann Hypothesis下，研究了zeta函数及其对数导数的加权联合二阶矩，为距临界线$a/\log T$处的该二阶矩提供了上下界。

Multiple Dedekind Zeta Functions

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are int…

迭代积分与上同调

Iterated integrals and cohomology

本研究受Manin非交换模符号与伪theta函数研究启发，提出标准上同调的扩展形式，既为相关迭代积分提供上同调解释，也给出多重zeta值关系的模对应物。

卷积多重zeta值若干变体的显式关系

Explicit Relations of Some Variants of Convoluted Multiple Zeta Values

本文建立卷积多重zeta值及其变体的显式关系，利用迭代积分推导公式，定义二级类比并推广积分-级数恒等式至多重polylogarithm和多重混合值，最后通过图形表示研究多重积分。