The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, $ξ(s)$. Thus, if the Riemann Hypothesis is true for the zeta-function, it is true for $ξ(s)$. Since $ξ(s)$ is entire, the zeros of $ξ'(s)$, its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of $ξ'(s)$ under the assumption that the Riemann Hypothesis is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros that are simple.
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.
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 $k \geq 0$.
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 $1/2-(\log\log T)^2/\log T<\Re s<1/2+(\log\log T)^2/\log T$.
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 $\mathrm{Re}\,s=1$.
Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, 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, Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. In this paper, we prove similar results related to the first derivative of the Dirichlet $L$-functions associated with primi…
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.
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.
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.
Analysis of Voros criterion: what derivatives involving the logarithm of the Riemann xi-function at z=1/2 should be non-negative for the Riemann hypothesis holds true
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.
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.
On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces
W. Luo研究了与紧致双曲黎曼曲面相关的Selberg zeta函数导数的零点分布。实质上，罗文的主要结果包括以下三点：临界线左边半平面中零点个数的渐近性;测量零点垂直分布的计数函数的渐近展开;测量零点到临界线水平距离的计数函数的渐近展开。在本文中，我们研究了更复杂的设置的Selberg zeta函数的导数的零点分布的非紧，有限体积双曲黎曼曲面。有许多困难存在于非紧的情况下，不存在于紧凑的设置，开始的事实，即在非紧的情况下，塞尔伯格zeta函数不满足类似的黎曼假设。更具体地说，我们实际上研究了ZH的导数的零点，其中Z是Selberg zeta函数，H是散射矩阵的狄利克雷级数分量，两者都与任意有限体积双曲黎曼曲面相关联。我们的主要结果地址有限的零在半平面的左边的临界线，零的垂直分布的渐近计数，和零的水平距离的渐近计数。
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…
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.
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…
Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $ζ'/ζ$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to Montgomery's pair correlation conjecture. This extends a result of Goldston, Gonek, and Montgomery where the mean value of $ζ'/ζ$ was considered.
The integral $R(t)=π^{-1}(lnζ(\frac{1}{2}+it)+i\vartheta (t))$ of the logarithmic derivative of the Hardy Z function $Z(t)=e^{i\vartheta (t)}ζ(\frac{1}{2}+it)$, where $\vartheta (t)$ is the Riemann-Siegel theta function, and $ζ(t)$ is the Riemann zeta function, is used as a basis for the construction of a pair of transcendental entire functions $ν(t)=-ν(1-t)=-{ΔR(\frac{i}{2}-it)}^{-1}=-G(\frac{i}{2}-it)$ where $G=-(ΔR(t))^{-1}$ is the derivative of the additive inverse of the reciprocal of the Laplacian of $R(t)$ and $χ(t)=-χ(1-t)=\dotν (t)=-iH(\frac{i}{2}-it)$ where $H(t)=\dot{G} (t)$ has roo…
Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the $ζ$-function to be a Lehmer pair in terms of derivatives of Hardy's $Z$-function. Furthermore, we connect Newman's conjecture with stationary points of the $Z$-function, and present some numerical results.
We derive a precise asymptotic expansion of the complete Kähler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. Futhermore, we use the modular form and Schwarzian derivative to explicitly determine the coefficients in the expansion of the complete Kähler-Einstein metric for punctured Riemann sphere with $3, 4, 6$ or $12$ omitting points.
Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not equal to 1/2 are non-negative if and only if the Riemann hypothesis holds true, and proved the relation k_n,a=n*(1-2a)/(n-1)!*d^n/dz^n((z-a)^(n-1)*ln(ξ(z))) taken at z=1-a. Assuming that the function /zeta(s) is non-vanishing for Re(s)>1/2+/Delta, where real 0</Delta<1/2, using this relation together with the functional equation for the /xi-function and the …
Update: This result was obtained by Milinovich with a better error term. He used $ζ'(s)$, but we considered $Z'(t)$. We corrected a typo in the main theorem. We consider the sum of the square of the derivative of Hardy's $Z$-function over the zeros of Hardy's $Z$-function. If the Riemann Hypothesis is true, it is equal to the sum of $|ζ'(ρ)|^2$, where $ρ$ runs over the zeros of the Riemann zeta-function. In 1984, Gonek obtained an asymptotic formula for the sum. In this paper we prove a sharper formula.
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…
It is known that we can construct the meromorphic function $Z_k(s)$ associated with the higher derivative of Hardy's $Z$-function. In this paper, we introduce the entire function derived from $Z_k(s)$, a generalisation of the Riemann $ξ$-function and prove some properties.
In this paper, we investigate the derivatives of L-functions, in particular, the Riemann zeta function, the Hasse-Weil L-function, the Rankin L-function and the Artin L-function, and survey the relations between the derivatives of L-functions and the geometry and arithmetic of the associated Shimura varieties.
We show that the density functions of nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which is appeared in value distributions of the logarithmic derivative of the Riemann zeta-function on vertical lines.
