Bounding the log-derivative of the zeta-function

Fourier Series Of the Derivatives of Hurwitz and Lerch Zeta Functions

As a function of second variable, we identify the Fourier series of Hurwitz zeta function and its derivatives on the unit interval. Consequently, we obtain results based on the formula for Fourier coefficients and also on Parseval's theorem. We do likewise in the case of Lerch's zeta function and its derivatives.

Lower bounds for moments of the derivative of the Riemann zeta function

We establish in this paper sharp lower bounds for the $2k$-th moment of the derivative of the Riemann zeta function on the critical line for all real $k \geq 0$.

On the zeros of the second derivative of the Riemann zeta function under the Riemann hypothesis

The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we extend the results of Akatsuka to the second derivative of the Riemann zeta function.

Numerical study of the derivative of the Riemann zeta function at zeros

The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.

Functional Equations of Tornheim Zeta Function and the alpha-calculus of r-th order derivative of Zeta(s,alpha)

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the tool,we obtain with ease an expression for T(s,s,s) for any complex s;and evaluate T(n,n,n) for any integer n>=1;and also evaluate T(n1,0,n2)for integers n1,n2>1 with n1+n2 odd.We obtain for Tornheim zeta function,the counterparts of Euler's formula for zeta(2n)and its analogue for zeta(2n+1),where Zeta(s) is Riemann zeta function.

Omega-theorems for the Riemann zeta function and its derivatives near the line $\mathrm{Re}\,s=1$

We introduce a generalization of the method of S. P. Zaitsev. This generalization allows us to prove omega-theorems for the Riemann zeta function and its derivatives in some regions near the line $\mathrm{Re}\,s=1$.

Double shuffle relations for q-analogues of multiple zeta values, their derivatives and the connection to multiple Eisenstein series

We study a certain class of q-analogues of multiple zeta values, which appear in the Fourier expansion of multiple Eisenstein series. Studying their algebraic structure and their derivatives we propose conjectured explicit formulas for the derivatives of double and triple Eisenstein series.

混合导数的黎曼ζ函数离散二次矩

The discrete second moment of mixed derivatives of the Riemann zeta function

本文研究了黎曼ζ函数在ζ零点处混合导数的离散二次矩的渐进行为，扩展了Conrey--Snaith和Milinovich的工作，提供了无条件和有条件误差项。

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.

On the $a$-points of the derivatives of the Riemann zeta function

We prove three results on the $a$-points of the derivatives of the Riemann zeta function. The first result is a formula of the Riemann-von Mangoldt type; we estimate the number of the $a$-points of the derivatives of the Riemann zeta function. The second result is on certain exponential sum involving $a$-points. The third result is an analogue of the zero density theorem. We count the $a$-points of the derivatives of the Riemann zeta function in $1/2-(\log\log T)^2/\log T<\Re s<1/2+(\log\log T)^2/\log T$.

A note on the zeros of zeta and $L$-functions

Let $πS(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound for a large class of $L$-functions including those which arise from cuspidal automorphic representations of GL($m$) over a number field. We also prove a number of related results including bounding the order of vanishing of an $L$-function at the central point and bounding the height of the lowest zero of an $L$-function.

寻找交替ζ函数导数之间的近似多项式依赖

In Search of Approximate Polynomial Dependencies Among the Derivatives of the Alternating Zeta Function

本文通过数值证据探讨交替ζ函数及其导数之间的近似多项式依赖关系，并提出相关猜想。

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.

黎曼zeta函数导数的复矩

Complex moments of the derivative of the Riemann zeta function

本文通过两种随机矩阵方法（Selberg积分精确公式与混合方法）猜想黎曼zeta函数导数在非平凡零点处的复矩结果。

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.

Precise Definition And Analyticity of r-th order partial derivative of zeta(s,alpha)

S.Ramanujan[1] was aware of power series expression in alpha of zeta(s,1-alpha)for complex s and for 0<=alpha<1,which he did not explore very far.Author[2] had derived power series expression in alpha of zeta(s,1+alpha)for complex α with |alpha|<1 and had shown in author[3] that the power series of zeta(-n,alpha) for integral n>=0,is a polynomial in α .On this backdrop, we give here the precise definition of r-th order partial derivative w.r.t s of zeta(s,alpha) in terms of power series in alpha for complex s and alpha . We also discuss the analyticity of r-th order partial derivative w.r.t s …

On the zeros of the $k$-th derivative of the Riemann zeta function under the Riemann hypothesis

The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we generalize the results of Akatsuka to the $k$-th derivative (for positive integer $k$) of the Riemann zeta function.

The alpha-calculus-cum-alpha-analysis of r-th order derivative of zeta(s,alpha)

For the higher order derivative(with respect to the first variable) of Hurwitz zeta function,we discuss as a function of the second variable,the location and the nature of its singularities and obtain the formulae for its derivative and primitive and also discuss its Riemann integrability on the unit interval and on the interval[1,infinity).We also evaluate certain integrals on the unit interval involving these functions.

Riemann zeta 函数导数的整数矩

Integer moments of the derivatives of the Riemann zeta function

本文猜想 Riemann zeta 函数在非平凡零点处乘积的渐近展开，通过位移求导得到混合导数的整数矩，推广了先前复矩结果，并分别用随机矩阵理论和比值猜想两种方法处理。

垂直齐次数列上Riemann zeta函数导数的大值

Large values of derivatives of the Riemann zeta function on vertical homogeneous progressions

该研究通过共振方法得到垂直齐次数列上Riemann zeta函数导数最大值的下界，证明实部σ在合适区间时离散与连续情形量级相近，推进了黎曼zeta函数的相关数论研究。

An upper bound for discrete moments of the derivative of the Riemann zeta-function

Assuming the Riemann hypothesis, we establish an upper bound for the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where $k$ is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating, and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments of the zeta-function on the critical line.

Zeros of the Lerch zeta-function and of its derivative for equal parameters

A. Speiser proved that the Riemann hypothesis is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function left of the critical line. His result has been extended by N. Levinson and H.L. Montgomery to the statement that the Riemann zeta-function and its derivative have approximately the same number of non-real zeros left of the critical line. We obtain the Levinson-Montgomery type result for the Lerch zeta-function with equal parameters. For the Lerch zeta-function, the analogue of the Riemann hypothesis is usually not true and its zeros usually are distributed…

On expansions involving the Riemann zeta function and its derivatives

By studying the spectral aspects of the fractional part function in a well-known separable Hilbert space, we show, among other things, a rational approximation of the Riemann zeta function and its derivatives valid on every vertical line in the right half-planes $\Re s> 1/2$ and $\Re s >0.$ Moreover, we provide some discussions and explicit computations related to the fractional part function.

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.

zeta-star 对应的导数函数

The derivative function of the zeta-star correspondence

本文研究 zeta-star 对应的右导数，证明其连续性、跳跃与维数性质，并给出 Hölder 指数与 Hausdorff 谱的线性结果。

A further generalisation of sums of higher derivatives of the Riemann Zeta Function

We prove an asymptotic for the sum of $ζ^{(n)} (ρ)X^ρ$ where $ζ^{(n)} (s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $ρ$ denotes a non-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height $T$, as $T \rightarrow \infty$. We also specify what the asymptotic formula becomes when $X$ is a positive integer, highlighting the differences in the asymptotic expansions as $X$ changes its arithmetic nature.

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。

黎曼ζ函数导数的负离散矩

Negative discrete moments of the derivative of the Riemann zeta-function

该研究在数论领域，推导黎曼ζ函数导数负离散矩在满密度零点子族上的条件上界，k≤1/2时2k阶矩界近乎最优，还在相关猜想下得到更大零点子族上同等强度的负矩上界。

Zeros of the extended Selberg class zeta-functions and of their derivatives

Levinson and Montgomery proved that the Riemann zeta-function $ζ(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $ζ'(1/2+it)=0$ implies $ζ(1/2+it)=0$. Here we obtain that in small areas located to the left of the critical line and near it the functions $ζ(s)$ and $ζ'(s)$ have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class $S$. We also consider zero trajectories of a certain family of zeta-functions from $S$.

On the higher derivatives of Z(t) associated with the Riemann Zeta-Function

Let $Z(t)$ be the classical Hardy function in the theory of the Riemann zeta-function. The main result in this paper is that if the Riemann hypothesis is true then for any positive integer $n$ there exists a $t_{n}>0$ such that for $t>t_{n}$ the function $Z^{(n+1)}(t)$ has exactly one zero between consecutive zeros of $Z^{(n)}(t)$.

黎曼zeta函数高阶导数的离散均值定理

A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

通过给出黎曼zeta函数非平凡零点上导数和的完全渐近展开，证明了其n阶导数的均值在n为奇数时为正、偶数时为负。

Fractional differential relations for the Lerch zeta function

Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch 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。

广义多重 Hurwitz zeta 函数在非正整数处的值与导数值

Values and derivative values at nonpositive integers of generalized multiple Hurwitz zeta functions

本文建立广义 Hurwitz 型多重 zeta 函数的亚纯延拓，导出其沿给定方向在非正整数处的值及导数值公式，并应用于 Witten zeta 函数与 $\mathfrak{g}_2$ 表示数的渐近分析。

Zeros of high derivatives of the Riemann Zeta function

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of the k-th derivative of the zeta function, we describe a unexpected phenomenon, which implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these n…

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $ζ(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

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。

The distribution of $k$-free numbers and the derivative of the Riemann zeta-function

Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $ζ(s)$. Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}μ_k(n)-\frac{x}{ζ(k)}$ and $μ_k(n)$ is the characteristic function of $k$-free numbers. Finally, we make a conjecture about the maximum order of $M_k(x)$ by heuristic analysis on the tail of the limiting distribution.

On the distribution of the zeros of the derivative of the Riemann zeta-function

We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\Re(s)=σ$ satisfying $(\log T)^{-1/3+ε} \leq (2σ-1) \leq (\log \log T)^{-2}$, we show that the number of zeros of $ζ'(s)$ with imaginary part between zero and $T$ and real part larger than $σ$ is asymptotic to $T/(2π(σ-1/2))$ as $T \rightarrow \infty$. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for $σ$ in this range the zeros of $ζ'(s)$ are horizontally distributed like the zeros of the derivative of chara…

黎曼zeta函数导数的矩：特征多项式与混合公式

Moments of derivatives of the Riemann zeta function: Characteristic polynomials and the hybrid formula

本文通过圆酉群特征多项式导数矩的渐近式与混合模型两种途径，提出黎曼zeta函数混合导数在非平凡零点处取值的矩猜想。

环形数域 Dedekind ζ 函数导数的极值

Extreme values of derivatives of the Dedekind zeta function of a cyclotomic field

本文通过双版本卷积公式和共振方法，研究环形数域 Dedekind ζ 函数导数的极值，并推广了 Yang 在 2022 年的工作。

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-p…

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.

Computation of the derivative of the Hurwitz zeta-function and the higher Kinkelin constants

I use the numerical values of the generalised Glaisher-Kinkelin-Bendersky (GKB) constants to give numerical values for the derivatives of the Hurwitz zeta function at negative integers, rather than the other way round. I point out that both Glaisher's numerical approach and Bendersky's recursion for the generalised gamma function were anticipated by Jeffery in 1862 who gave the value of the second constant as an example. I therefore propose that GKB become GKBJ.

黎曼ζ函数高阶导数的零点

Zeros of higher derivatives of Riemann zeta function

该研究将Conrey关于黎曼ζ函数的结果推广到高阶导数的更短区间，通过优化短区间内的mollifier长度改进了零点密度估计的经典误差项，并证明Matsumoto-Tanigawa的$η_k$-函数几乎所有非平凡零点聚集在临界线附近。

On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces

W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness for the number of zeros in the half plane to the left of the critical line; an asymptotic expansion for the counting function measuring the vertical distribution of zeros; and an asymptotic expansion for the counting function measuring the horizontal distance of zeros from the critical line. In the present article, we study the more complicated setting of d…

黎曼 zeta 函数导数离散均值的一个启发式方法

A heuristic for discrete mean values of the derivative of the Riemann zeta function

本文介绍了 Shanks 关于黎曼 zeta 函数导数均值猜想的推广历史，并提出一个简单的启发式方法，用于确定其高阶导数在非平凡零点上平均值渐近公式的主项及符号。

On an inequality for the Riemann zeta-function in the critical strip

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |ζ(1-s)| <= |ζ(s)| in the strip 0< Re s<1/2,\ |\Im s| >= 12. Moreover, we establish a sufficient condition of the validity of the Riemann hypothesis in terms of the derivative with respect to Re s of |ζ(s)|^2 and conjecture its necessity.

Roots of the derivative of the Riemann zeta function and of characteristic polynomials

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.

Extreme values of derivatives of the Riemann zeta function

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|ζ^{(\ell)}\Big(1+it\Big)\right| \geqslant e^γ\cdot \ell^{\ell}\cdot (\ell+1)^{ -(\ell+1)}\cdot\Big(\log_2 T - \log_3 T + O(1)\Big)^{\ell+1} \,, \end{equation*} where $γ$ is the Euler constant. We also establish lower bounds for maximum of $\big|ζ^{(\ell)}(σ+it)\big|$ when $\ell \in \mathbb N $ and $σ\in [1/2, \,1)$ are fixed.

Moments of the derivative of the Riemann zeta-function and of characteristic polynomials

We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann zeta-function on the critical line. We do the same for the analogue of Hardy's Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.

交替 Hurwitz zeta 函数及其导数的渐近展开

Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives

本文研究交替 Hurwitz zeta 函数，建立了其在参数模趋于无穷时的渐近展开及余项估计，并进一步给出其高阶导数渐近展开和一个新的精确级数表示。

Meromorphicity of some deformed multivariable zeta functions for $F_1$-schemes

Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivati…

Representations for the derivative at zero and finite parts of the Barnes zeta function

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.