David W. Farmer
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.
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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.
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.
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.
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.
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.
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。
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…
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。
We prove and conjecture several relations between multizeta values for $\mathbb{F}_q[t]$, focusing on zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or equivalently algebraic). In particular, we describe them conjecturally fully for $q=2$, or more generally for any $q$ for `even' weight (`eulerian' tuples). We provide some data in support of the guesses.
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。
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.
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。
Omega Theorems for Logarithmic Derivatives of Zeta and L-functions Near the 1-line
本文利用共振方法证明了 Riemann zeta 函数在 1 线附近对数导数的 omega 定理，给出了特定不等式解的存在性与测度条件性下界，并将结果推广至 Dirichlet L 函数。
On the value distribution of the Riemann zeta-function with general shifts
研究一般位移下黎曼ζ函数的值分布，通过建立偏差估计、证明普遍性定理及获得定量定理，在对数幂位移方面取得成果，此前一般位移普遍性结果未涵盖。
Based on the notion of Stark units we present a new approach that obtains refinements of log-algebraic identities for Anderson t-modules. As a consequence, we establish a generalization of Chang's theorem on logarithmic interpretations for special characteristic p multiple zeta values (MZV's) and recover many earlier results in this direction. Further, we devise a direct and conceptual way to get logarithmic interpretations for both MZV's and v-adic MZV's. This generalizes completely the work of Anderson and Thakur for Carlitz zeta values.
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.
We develop a method for mean-value estimation of long Dirichlet polynomials. For an application, we use our method to study properties of the logarithmic derivative of the Riemann zeta function.
We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected…
Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta function
本文通过三角多项式论证，改进了黎曼ζ函数在接近1处的对数导数和倒数的显式界，并在经典无零区域和科罗波夫-文诺格拉夫无零区域中取得了新的无条件界。
A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function
介绍高斯 - 佩龙素数力缺陷，将显式公式转化为零几何局部诊断，给出误差函数素数权重等，证明零侧公式等，在相关假设下使缺陷局部化，为黎曼ζ函数零几何提供局部诊断。
We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to Nörlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values …
In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $ζ(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an asymptotic formula for the central derivatives $ζ^{(2n)}(1/2)$ that is accurate to all orders, which allows us to prove the hyperbolicity of a density $1$ subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all $d\leq 8$. These results follow from a general theorem which models such polynomials by Hermite polynomials. In…
Quadratic enrichment of the logarithmic derivative of the zeta function
本文将有限域上代数簇的 zeta 函数对数导数富集到 Grothendieck–Witt 群系数的幂级数，证明其与实点拓扑的联系，并对胞腔概形建立有理性结果，验证其为母题测度。
Cauchy laws for zeta logarithmic derivatives and stationary Stieltjes transforms
本文证明Riemann zeta函数零点及有限域簇zeta函数对数导数的Cauchy极限定律，给出显式误差界，并建立平稳点测度Stieltjes变换的收敛定理。
Bounding zeta on the 1-line under the partial Riemann hypothesis
本文假设Riemann假设直至高度$T$成立，给出了Riemann zeta-函数在$\Re{s}=1$直线上的显式界，并改进了对数导数及其倒数在有限区域内的界。
On the series expansion of the prime zeta function about $s=1$ and its coefficients
本文推导了素数ζ函数在s=1处的级数展开，并证明了其系数的通用公式，类似于黎曼ζ函数的Stieltjes系数。结果可视为对梅特恩斯定理的推广，并通过数值验证计算了高精度公式。
We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero $ρ$ (trivial or nontrivial) and each zeta fixed point $ψ$ there is a nearly logarithmic spiral $s_{ρ, ψ}$ with center $ψ$ containing $ρ$. (2) $s_{ρ, ψ}$ interpolates a subset $B_{ρ, ψ}$ of the backward zeta orbit of $ρ$ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, $ζ(B_{ρ, ψ}) = B_{ρ, ψ} \cup \{0 \}$. (4) $B_{ρ, ψ}$ has nearly uniform angular distribution around the center of $s_{ρ, ψ}$. We will make these statements precise.
We establish new explicit zero-free regions for the Dedekind zeta-function. Two key elements of our proof are a non-negative, even, trigonometric polynomial and explicit upper bounds for the explicit formula of the so-called differenced logarithmic derivative of the Dedekind zeta-function. The improvements we establish over the last result of this kind come from two sources. First, our computations use a polynomial which has been optimised by simulated annealing for a similar problem. Second, we establish sharper upper bounds for the aforementioned explicit formula.
For each positive characteristic multiple zeta value (defined by Thakur), the first and third authors constructed a $t$-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and $t$-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.
$Ω$-results for the logarithmic derivative of $ζ(s)$ on vertical homogeneous progressions
本文在垂直齐次等差数列上建立黎曼ζ函数对数导数的$Ω$-结果，表明离散情形与连续情形数量级相似。
On convergence of points to limiting processes, with an application to zeta zeros
本文研究实轴上随机平移点序列的多种极限点过程收敛概念的等价性，证明相关Tauberian定理，并将结果应用于Riemann zeta函数零点，证明GUE假设的多种表述等价。
Riemann and the logarithmic derivatives of zeta
本文基于 Riemann 遗稿，给出了 ζ(s) 在 s=1/2 处一阶和二阶对数导数的显式公式，并解释了其推导过程。
In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein constructed a simple (non-holomorphic) completion of this form which is doubly periodic. This theorem has begun to play an important role in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as well as Alfes, Griffin, Ono, and the author. In particular, this simple completion of $ζ$ provides a powerful method to construct harmonic Maass…
We study some "density function" related to the value-distribution of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density function in a wide class of $L$-functions. We prove that certain mean values of $L$-functions in the class are represented as integrals involving the related density functions.
Dynamics of multiples of zeta by exponential functions
本文通过数值实验研究指数函数乘以zeta函数的动力学，发现其不动点以对数模式趋近非平凡零点，并提出了相关猜想与条件性证明。
Pair correlation for sums of two ordinates of zeros of the Riemann zeta function
在黎曼假设下，扩展蒙哥马利对关联方法研究黎曼ζ函数非平凡零点两纵坐标之和的差分布，证明相关对关联函数表达式，指出两纵坐标之和无能级排斥，其两点关联函数如泊松过程恒为1。
Explicit conditional bounds for $ζ(s)$ at the edge of the critical strip
本文在黎曼假设下，利用Guinand-Weil显式公式和极值带限主次量，获得ζ(s)在Re s=1线上对数导数实部的显式界，并改进了Lamzouri等人和Chirre等人关于模长界的估计。
We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the family of functions $V_z: s \mapsto ζ(s) \exp (zs)$. We observe convergence of $V_z$ fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.
We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann $ζ$ function at odd integers are irrational. These generalizations concern multiple series of hypergeometric type, which can be written as linear forms in some specific multiple zeta values. The proof makes use of the regularization procedure for multiple zeta values with logarithmic divergence.
Littlewood's estimates for $L$-functions in the hyperelliptic ensemble
本文研究函数域上二次 Dirichlet $L$-函数的 Littlewood 估计类似物，在函数域中建立了比数域更精细的结果，并据此对 Riemann zeta-函数提出猜测，方法上依赖 Fourier 分析。
In this article, we develop a formula for an inverse Riemann zeta function such that for $w=ζ(s)$ we have $s=ζ^{-1}(w)$ for real and complex domains $s$ and $w$. The presented work is based on extending the analytical recurrence formulas for trivial and non-trivial zeros as to solve an equation $ζ(s)-w=0$ for a given $w$-domain using logarithmic differentiation and zeta recursive root extraction methods. We further explore formulas for trivial and non-trivial zeros of the Riemann zeta function in greater detail, and next, we also explore an expansion of the inverse zeta function by an attracto…
In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number.
In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as (reciprocal) logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms are given for Euler's constant and the constant ln(2*pi), and yet another generalization of Euler's constant is proposed and various formulas for …
Let $\CmZ_{Y_0(N)}$ be the constant term of the logarithmic derivative at $s=1$ of the Selberg zeta function of the modular curve $Y_0(N)$. Jorgenson and Kramer established the bound $\CmZ_{Y_0(N)}=O_ε(N^ε)$, $ε>0$ by relating it to geometric invariants. In this article we give, for $N$ prime, another proof via $L$-functions and exponential sums improving on a previous approach by Abbes-Ullmo and Michel-Ullmo. We further derive a power of $\log N$ bound along the same line.
The error term of the fourth moment of the Riemann zeta-function
本文改进了黎曼zeta函数四次矩误差项及其平方积分的对数幂次界，从而对8至12次矩获得小的对数改进，尤其对12次幂矩有适度提升。
The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of complicated integrals involving logarithmic functions. This work deals with a systematic review of logarithmic integrals starting from Malmsten integrals to classical collection of Integrals, Series and Products by I. S. Gradshteyn and I. M. Ryzhik [1] to recent times. The evaluation of these types of integrals involves higher transcendental functions (i.e., Hur…
We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli-Carlitz fractions and power sums as well as their multi-variable generalizations. Resulting understanding of their factorizations has arithmetic significance, as w…
Laurent series expansions for the Barnes multiple zeta function
本文研究Barnes多重zeta函数的Laurent系数，在最高极点给出显式极限公式，在较低极点建立与Taylor系数的全阶关系，并利用整函数性质分析大阶行为，推广了常数项的有限部分表示。
Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control. The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor. Practical parameter choices are discussed.
We consider the problem whether the ordinates of the non-trivial zeros of $ζ(s)$ are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros $(x_n)$ are uniformly distributed modulo 1. Odlyzko conjectured this to be true. This is far from being proved, even assuming the Riemann hypothesis (RH, for short). Applying the Piatetski-Shapiro $11/12$ Theorem we are able to show that, for $0<κ<6/5$, the mean value $\frac1N\sum_{n\le N}\exp(2πi κx_n)$ tends to zero. The case $κ=1$ is especially interesting. In this case the Prime Number Theorem is sufficient to prove that…
It is shown that the absolute values of Riemann's zeta function and two related functions strictly decrease when the imaginary part of the argument is fixed to any number with absolute value at least 8 and the real part of the argument is negative and increases up to 0; extending this monotonicity to the increase of the real part up to 1/2 is shown to be equivalent to the Riemann Hypothesis. Another result is a double inequality relating the real parts of the logarithmic derivatives of the three functions under consideration.
We numerically study the statistical properties of differences of zeros of Riemann zeta function and L-functions predicted by the theory of the eñe product. In particular, this provides a simple algorithm that computes any non-real Riemann zeros from very large ones ("self-replicating property of Riemann zeros"). Also the algorithm computes the full sequence of non-real zeros of Riemann zeta function from the sequence of non-real zeros of any Dirichlet L-function ("zeros of L-functions know about Riemann zeros"). We also check that the first error to the convergence to the classical GUE statis…
Polylogarithmic Analogues of Euler's Constant
引入一族常数 C_m，作为欧拉常数的多对数类似物，研究其性质并建立与伽马函数相关的迭代对数积分表示，进一步引入多对数zeta势和多对数伽马函数，并应用于将C_m与涉及黎曼zeta函数的狄利克雷级数的特殊值联系起来。
In recent work, Fyodorov and Keating conjectured the maximum size of $|ζ(1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the characteristic polynomial of a random matrix; relating the random matrix problem to another problem from statistical mechanics; and applying a heuristic analysis of that problem. In this note we recover a conjecture like that of Fyodorov and Keating, but using a different model for $|ζ(1/2+it)|$ in terms of a random Euler product. In this case the probabilistic model reduces to studying the supremum of …
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。
Conditional estimates for the logarithmic derivative of Dirichlet $L$-functions
本文在广义黎曼猜想假设下，对模 q≥10^30 的本原 Dirichlet L-函数的对数导数建立了 q 方面的显式界，并在 σ=1 处改进了 Ihara-Murty-Shimura 的结果，同时给出 Riemann zeta 函数的类似估计。
A Regularised Wallis Hierarchy
通过将Wallis乘积因子提升到多项式权重，引入正则化Wallis乘积层级，并利用最小指数反项消除对数展开中的不可和项，得到收敛乘积，其对数表示为显式zeta函数尾部。
We initiate the study of spectral zeta functions $ζ_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $ζ(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of …
We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant, gamma.
The normalized approximation function for multiple zeta-star values
该研究受拉格朗日谱及其连分数倒数形式启发，引入多重zeta星值的归一化逼近函数，证明其正则性、可测性等性质，得出几乎所有大于1的实数对应函数值为0且像集稠密的结论。
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