Generalization of Okamoto's Equation to Arbitrary2×2Schlesinger System

The2×2Schlesinger system for the case of four regular singularities is equivalent to the Painlevé VI equation. The Painlevé VI equation can in turn be rewritten in the symmetric form of Okamoto's equation; the dependent variable in Okamoto's form of the PVI equation is the (slightly transformed) logarithmic derivative of the Jimbo-Miwa tau-function of the Schlesinger system. The goal of this note is twofold. First, we find a universal formulation of an arbitrary Schlesinger system with regular singularities in terms of appropriately defined Virasoro generators. Second, we find analogues of Okamoto's equation for the case of the2×2Schlesinger system with an arbitrary number of poles. A new set of scalar equations for the logarithmic derivatives of the Jimbo-Miwa tau-function is derived in terms of generators of the Virasoro algebra; these generators are expressed in terms of derivatives with respect to singularities of the Schlesinger system.

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OMEGA THEOREMS FOR $\frac{L'}{L}(1, \chi_D)$

International Journal of Number Theory ◽

10.1142/s1793042112501485 ◽

2013 ◽

Vol 09 (03) ◽

pp. 561-581 ◽

Cited By ~ 4

Author(s):

M. MOURTADA ◽

V. KUMAR MURTY

Keyword(s):

Classical Result ◽

London Math ◽

Quadratic Field ◽

Analogous Result ◽

Logarithmic Derivative ◽

Dirichlet Character ◽

Derivative Formula ◽

Derivatives Of ◽

Logarithmic Derivatives

A classical result of Chowla [Improvement of a theorem of Linnik and Walfisz, Proc. London Math. Soc. (2) 50 (1949) 423–429 and The Collected Papers of Sarvadaman Chowla, Vol. 2 (Centre de Recherches Mathematiques, 1999), pp. 696–702] states that for infinitely many fundamental discriminants D we have [Formula: see text] where χD is the quadratic Dirichlet character of conductor D. In this paper, we prove an analogous result for the logarithmic derivative [Formula: see text], and investigate the growth of the logarithmic derivatives of real Dirichlet L-functions. We show that there are infinitely many fundamental discriminants D (both positive and negative) such that [Formula: see text] and infinitely many fundamental discriminants 0 < D such that [Formula: see text] In particular, we show that the Euler–Kronecker constant γK of a quadratic field K satisfies γK = Ω( log log |dK|). We get sharper results assuming the GRH. Moreover, we evaluate the moments of [Formula: see text].

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New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0212 ◽

2008 ◽

Vol 464 (2091) ◽

pp. 711-731 ◽

Cited By ~ 4

Author(s):

Mark W Coffey

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Logarithmic Derivative ◽

Bernoulli Numbers ◽

Stieltjes Constants ◽

The Riemann Zeta Function ◽

Power Series Expansions ◽

Riemann Zeta ◽

Derivatives Of ◽

Logarithmic Derivatives

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of λ k is developed in terms of the Stieltjes constants γ j and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we find a new representation of the constants η j entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about s =1. We also demonstrate that the η j coefficients are expressible in terms of the Bernoulli numbers and certain other constants. We determine new properties of η j and σ j , where are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function.

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COMPLETELY MONOTONIC FUNCTIONS RELATED TO LOGARITHMIC DERIVATIVES OF ENTIRE FUNCTIONS

Analysis and Applications ◽

10.1142/s0219530511001911 ◽

2011 ◽

Vol 09 (04) ◽

pp. 409-419 ◽

Cited By ~ 4

Author(s):

HENRIK L. PEDERSEN

Keyword(s):

Entire Functions ◽

Logarithmic Derivative ◽

Psi Function ◽

Stieltjes Function ◽

Gamma Functions ◽

Monotonic Functions ◽

Completely Monotonic ◽

Completely Monotonic Functions ◽

Derivatives Of ◽

Logarithmic Derivatives

The logarithmic derivative l(x) of an entire function of genus p and having only non-positive zeros is represented in terms of a Stieltjes function. As a consequence, (-1)p(xml(x))(m+p) is a completely monotonic function for all m ≥ 0. This generalizes earlier results on complete monotonicity of functions related to Euler's psi-function. Applications to Barnes' multiple gamma functions are given.

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Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ac3f88 ◽

2021 ◽

Author(s):

Bruno Carneiro da Cunha ◽

Tiago Anselmo da Silva ◽

Rhodri Nelson ◽

Darren Crowdy ◽

Salman Abarghouei Nejad

Keyword(s):

Conformal Map ◽

Constructive Approach ◽

Tau Function ◽

Upper Half Plane ◽

Fuchsian Equation ◽

Simply Connected ◽

Derivatives Of ◽

Logarithmic Derivatives ◽

Theoretical Results

Abstract The prevertices of the conformal map between a generic, n-vertex, simply connected, polycircular arc domain and the upper half plane are determined by ﬁnding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of 5 vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko & Lisovyy [Comm. Math. Phys., 363, 2018)] is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.

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SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

Journal of the Australian Mathematical Society ◽

10.1017/s1446788710000029 ◽

2010 ◽

Vol 88 (2) ◽

pp. 145-167 ◽

Cited By ~ 10

Author(s):

I. CHYZHYKOV ◽

J. HEITTOKANGAS ◽

J. RÄTTYÄ

Keyword(s):

Differential Equations ◽

Unit Disc ◽

Logarithmic Derivative ◽

Maximum Modulus ◽

Exceptional Set ◽

Proximate Order ◽

Upper Density ◽

Linear Differential ◽

Logarithmic Derivatives ◽

Growth Of Solutions

AbstractNew estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

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Logarithmic Derivatives of Heat Kernels and Logarithmic Sobolev Inequalities with Unbounded Diffusion Coefficients on Loop Spaces

Journal of Functional Analysis ◽

10.1006/jfan.2000.3592 ◽

2000 ◽

Vol 174 (2) ◽

pp. 430-477 ◽

Cited By ~ 12

Author(s):

Shigeki Aida

Keyword(s):

Diffusion Coefficients ◽

Heat Kernels ◽

Sobolev Inequalities ◽

Loop Spaces ◽

Logarithmic Sobolev Inequalities ◽

Derivatives Of ◽

Logarithmic Derivatives

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Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane

Matematychni Studii ◽

10.30970/ms.54.2.172-187 ◽

2020 ◽

Vol 54 (2) ◽

pp. 172-187

Author(s):

I.E. Chyzhykov ◽

A.Z. Mokhon'ko

Keyword(s):

Meromorphic Function ◽

Finite Order ◽

Half Plane ◽

Meromorphic Functions ◽

Logarithmic Derivative ◽

Sharp Estimates ◽

Exceptional Sets ◽

Upper Half Plane ◽

Logarithmic Derivatives

We established new sharp estimates outside exceptional sets for of the logarithmic derivatives $\frac{d^ {k} \log f(z)}{dz^k}$ and its generalization $\frac{f^{(k)}(z)}{f^{(j)}(z)}$, where $f$ is a meromorphic function $f$ in the upper half-plane, $k>j\ge0$ are integers. These estimates improve known estimates due to the second author in the class of meromorphic functions of finite order.Examples show that size of exceptional sets are best possible in some sense.

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On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1993-007-1 ◽

1993 ◽

Vol 36 (1) ◽

pp. 38-44

Author(s):

Alan D. Gluchoff

Keyword(s):

Integral Means ◽

Koebe Function ◽

Starlike Mappings ◽

Derivatives Of ◽

Logarithmic Derivatives

AbstractThe purpose of this paper is to prove some facts about integral means of (d2/dz2)(log[f(z)/z])—or equivalently f″/f, for f in a class of starlike mappings of a "singular" nature. In particular it is noted that the Koebe function is not extremal for the Hardy means Mp(r,f″/f) for functions in this class.

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