We establish Connes's local trace formula
(related to the explicit formulae of number theory) for the quaternions. This is done as an application of a study of the central operator H = log(|x|) + log(|y|) in the context of invariant harmonic analysis. The multiplicative analysis of the additive Fourier transform gives a spectral interpretation to generalized ``Tate Gamma functions'' (closely akin to the Godement-Jacquet ``γ(s,π,ψ)'' functions.) The analysis of H leads furthermore to a spectral interpretation for the logarithmic derivatives of these Gamma functions (which are involved in ``explicit formulae''.)
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.
Faulhuber and Steinerberger conjectured that the logarithmic derivative of $\vartheta_4$ has the property that $y^2\,\vartheta_4'(y)/\vartheta_4(y)$ is strictly decreasing and strictly convex. In this small note, we prove this conjecture.
We construct (generalized) logarithmic derivatives for general n-dimensional local fields K of mixed characteristics (0,p) in which p is not necessarily a prime element with residue field k such that [k:k^p]=p^{n-1}. For the construction of the logarithmic derivative map, we define n-dimensional rings of overconvergent series and show that - as in the 1-dimensional case - they can be interpreted as functions converging on some annulus of the open unit p-adic disc. Using the generalized logarithmic derivative, we give a new construction of Kato's n-dimensional dual exponential map.
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.
Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications(Ministry of Education), Tongji University（同济大学数学科学学院，智能计算与应用教育部重点实验室）
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 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.
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.
In this work we continue to study the properties of polynomials of binomial type and their canonical continuations to the complex index by exploring the properties of transformation T:=1/dlog which acts on formal power series $f(x)$ of the form $x+x^2\mathbb{C}[[x]]$.
In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r-1) for r > 1/2. This continued fraction is a generalization of the Brouncker's continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at the infinity are also studied. The generalizations of some Ramanujan's formulas are presented.
In this work we introduce interesting infinite series, related to Ramanujan-Soldner constant. Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator 1/dlog, acting on formal power series. In addition, several properties of polynomials of binomial type associated to elementary functions are discussed.
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…
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.
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.
Let $d\in\mathbb{N}$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,π\otimesχ_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $χ_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}…
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.
This work computes the asymptotics of the non-integer moments of the logarithmic derivative of characteristic polynomials of matrices from the $SO(2N+1)$ ensemble. It follows from work of Alvarez and Snaith who computed the asymptotics of the integer moments of the same statistic over both $SO(N)$ ensembles as well as the $USp(2N)$ ensemble.
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…
We study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices. In particular, we compute the asymptotics for large matrix size, $N$, of these moments evaluated at points which are approaching 1. This follows work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith where they compute these asymptotics in the case of unitary random matrices.
