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Topic: Stieltjes constants

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Riemann zeta function

Dirichlet series

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	Euler-Mascheroni constant - Wikipedia, the free encyclopedia
		The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory.
		Sometimes it is called simply the Euler constant, though it ought not to be confused with e, which is often called Euler's number.
		The Euler-Mascheroni constant was used in the solution to the Car Talk puzzler for the week of 23 October 2006.
	en.wikipedia.org /wiki/Euler-Mascheroni_constant (830 words)

	PrintThisPage
		Liang and J. Todd, The Stieltjes constants, J.
		Bi Cheng Yang and Kang Wu, An inequality for the Stieltjes constants (in Chinese), J.
		Kreminski, High-accuracy numerical estimates of Stieltjes constants and multiple zeta (Euler/Zagier) sums (1999).
	www.mathsoft.com /printThisPage.aspx?1218 (282 words)

	Math Games: Evil Numbers
		The first 144 numbers in the CF of the Gauss constant sum to 666.
		The first 174 numbers in the CF of the Landau Constant sum to 666.
		The farthest he had to reach for any constant was with Catalan's constant, which wasn't evil until he took every 28th digit.
	www.maa.org /editorial/mathgames/mathgames_10_04_04.html (882 words)

	[No title]
		Direct use of formula (1) to compute Euler constant is of poor interest since the convergence is very slow.
		But, a few years later, in 1809, Johann von Soldner (1766-1833) found a value of the constant which was in agreement only with the first 19 decimal places of Mascheroni's calculation...
		Integral and series formulae for the Euler constant can be found in Collection of formulae for the Euler constant.
	numbers.computation.free.fr /Constants/Gamma/gamma.html (1154 words)

	Citations: Recherches sur les fractions continues - Stieltjes (ResearchIndex)
		Stieltjes, Recherches sur les fractions continues, Annales de la Facult'e des Sciences de Toulouse 8 (1894), 1--122; 9 (1895), 5-47.
		In fact, from the point of view of J continued fractions obtained from the contraction of an S continued fraction with positive coecients, Stieltjes proved the existence of a positive linear functional such that the denominators of the approximants are orthogonal with respect to it [23, x11]....
		The crucial point in the calculation is the Z periodicity of sin(2 x) Chihara [6] and later Leipnik [15] gave the following family of solutions to (2.3) concentrated on countable sets.
	citeseer.ist.psu.edu /context/127074/0 (2950 words)

	[No title]
		[INVERSE SYMBOLIC CALCULATOR] BASE TABLE OF CONSTANTS %0 C0000 %0 C0000 TABLE OF CONSTANTS %0 C0000 by %0 C0000 Simon Plouffe %0 C0000 CECM %0 C0000 Centre for Experimental & Constructive Mathematics %0 C0000 Simon Fraser University, Burnaby BC CANADA %0 C0000 Here is a set of known constants to a precision of 1024 digits.
		There are more than 315 other tables to a total of 8 %0 C0000 million constants at this present date.
		%K C0083 The calculation of this constant is at the origin of the Pentium Bug.
	www.math.ucl.ac.be /~magnus/num1a/constants0.txt (8879 words)

	C:\Program Files\Wolfram Research\Mathematica\3.0\webstuff\webpg1\test1.html
		are often called generalized Euler constants, and were studied by Stieltjes; hence sometimes they are instead known as Stieltjes constants.
		The sign pattern for the first eight hundred constants is very orderly; here are their signs, in the form of a listing of the length of the "runs" of negatives and positives.
		Related structure appears in the growth of the generalized Stieltjes constants (i.e., up to normalization, the Laurent series coefficients for the Riemann-Hurwitz zeta function).
	www.tamu-commerce.edu /coas/math/FACULTY/KREMIN/stieltjes/stieltjestestpage.html (617 words)

	Number Theory Workshop - Abstract
		The Stieltjes constants [1] \zeta_k have been of interest for over a century, yet their detailed behavior remains under investigation.
		We have derived new relations amongst the values \zeta_k(a) that are the expansion coefficients in the Laurent series for the Hurwitz zeta function.
		In addition, we have developed a new integral relationship for the Hurwitz zeta function and one of its implications for the Stieltjes constants.
	www.mscs.dal.ca /~dilcher/abs.html (1564 words)

	A set of 'gamma' constants
		We say that a constant is a member of the "gamma" set if there is a formula
		There is an infinity of formulas whereby the difference between a pair of converging curves yields a constant.
		However, for divergent curves other than the euler constant curves, I have not found a formula which produces a nonzero constant.
	www.angelfire.com /trek/nfold/gamma.html (208 words)

	[No title]
		Additional formulas involving the Stieltjes constants are also derived.
		Because the Stieltjes constants appear in many formulas, the constants were evaluated freshly for this work.
		Formulas for the $\gamma\sb n$ are derived with new error bounds, and a tabulation of the constants is given from $n=0$ to 100." _________________________________________________________________ 96m:11070 11M06 (11-02) Laurin\v cikas, Antanas(LI-VILN) Limit theorems for the Riemann zeta-function.
	www.math.niu.edu /~rusin/known-math/99/zeta (3655 words)

	Stieltjes constants - Wikipedia, the free encyclopedia
		In mathematics, the Stieltjes constants are the numbers γ
		The Stieltjes constants can also be defined as the value of the limit
		One such expansion, in terms of the falling factorial, is given in the article on the Gauss-Kuzmin-Wirsing operator.
	en.wikipedia.org /wiki/Stieltjes_constants (118 words)

	Mathematics of Computation
		are known as Stieltjes, or generalized Euler, constants.
		such as 53/100, 1/2, etc.) suggest that published bounds on the growth of the Stieltjes constants can be much improved, and lead to several conjectures.
		Zhang Nan-Yue and K. Williams, Some results on the generalized Stieltjes constants, Analysis 14 (1994), 147-162.
	www.ams.org /mcom/2003-72-243/S0025-5718-02-01483-7/home.html (364 words)

	Abstracts 5(2002)
		Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages.
		Hadamard-type inequalities are derived for g-convex dominated maps.
		Sharp bounds of the Cebysev functional for the Stieltjes integrals similar to the Grüss one and applications for quadrature rules are given.
	rgmia.vu.edu.au /abstracts2002.html (861 words)

	RedTram News Search Engine \| News on Physics everywhere (Site not responding. Last check: 2007-09-20)
		New results on the Stieltjes constants: Asymptotic and exact evaluation.
		The Stieltjes constants are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s=1.
		Motivated by the analysis of Schr\"odinger operators with periodic potentials we consider the following abstract situation: Let $\Delta_X$ be the Laplacian on a non-compact Riemannian covering manifold $X$ with a discrete isometric group $\Gamma$ acting on it such that the
	www.redtram.com /catalogue/world/physics/20050624/7 (908 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		These objects are related via resolvents and Laplace Stieltjes transforms: \[ (\lambda - A)^{-1} = \int_0^\infty e^{-\lambda t} \Phi (dt).
		Again by (\ref{e:4.5}), \[ \Phi_3(t) = \int_0^t \Phi_2(t-s) V(ds) \quad \mbox{ on } X. Since $V(t)$ maps into $X_\wp$, the Stieltjes integral \(\int_0^t \Phi_2'(t-s) V(ds) \) exists and is a continuous function of $t$ in operator norm on $X$.
		The operators $V(t)$ map into the one-dimensional subspace spanned by $\delta_0 \times m$ and so do their Laplace Stieltjes transforms $F(\lambda)$ which consequently are compact.
	math.smsu.edu /~journal/sample1 (8709 words)

	Miscellaneous literature and links
		These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function.
		We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations.
		The analysis is motivated by the evaluation of determinants on spheres which are treated both by a direct expansion method and by regularised sums.
	secamlocal.ex.ac.uk /~mwatkins/zeta/physics8.htm (4131 words)

	Mathematics
		The Fibonacci Sequence Module M -- including a proof that given any integer m, infinitely many Fibonacci numbers are divisible by m.
		http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/ lists a bunch of constants, but even more are listed by
		On the Random Character of Fundamental Constant Expansions, by D. Bailey and R. Crandall
	mcraefamily.com /Links/InfoMathFacts.htm (315 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		From: math@che.freesurf.fr (Math) Subject: Re: Approximations of power series Date: Sat, 25 Sep 1999 18:47:44 GMT Newsgroups: sci.math Keywords: Stieltjes' expansion of zeta(s) near s=1 In article
		For better >approximations, see any analytic number theory text, or maybe even >any set of math tables that includes the zeta function.
		> Well there is the well known formula of Stieltjes: Zeta[ 1+x] = 1/x + gamma + sum((-1)^n (stieltjes(n) x^n/n!, n=1->oo) where stieltjes(n) is a family of constants for which there are several representations and gamma is Euler gamma constant.
	www.math.niu.edu /Papers/Rusin/known-math/99/zeta_stieltjes (181 words)

	Mathematics of Computation
		S. Ehrich, Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas, in W. Gautschi, G. Golub, G. Opfer (Eds.), Applications and Computation of Orthogonal Polynomials, Proc.
		H. Jung, Estimates for the first and second derivatives of the Stieltjes polynomials, J. Approx.
		F. Peherstorfer, On the Asymptotic Behaviour of Functions of the Second Kind and Stieltjes Polynomials and on the Gauss-Kronrod Quadrature Formulae, J. Approx.
	www.mathaware.org /mcom/2006-75-254/S0025-5718-05-01795-3/home.html (588 words)

	Publication : T05/108 (Site not responding. Last check: 2007-09-20)
		Coffey, M.W.: New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants, preprint (Jan. 2005); Toward verification of the Riemann Hypothesis: application of the Li criterion, Math.
		Matsuoka, Y.: A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function J.
		Shinshu Univ. 53 (1985), 81-82; A sequence associated with the zeros of the Riemann zeta function Tsukuba J. Math.
	www-spht.cea.fr /articles_k2/t05/108/biblio (449 words)

	www.myspace.com/riemannzeta
		I really enjoy being computed analytically for even N using either contour integration or Parseval's theorem with the appropriate Fourier series.
		(An unexpected and important formula involving the product of primes was first discovered by Euler in 1737!) I also sometimes like to be expressed analytically in terms of,"PI, Theta (N)", the Euler-Mascheroni constant Y, and the Stieltjes constants YsubI.
		I'm defined over the complex plane for one complex variable, and am conventionally denoted "s" (instead of the usual "z") in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859).
	www.myspace.com /riemannzeta (262 words)

	Power Series Expansions of Riemann's Zeta Function -- from Mathematica Information Center
		We show how high-precision values of the coefficients of power series expansions of functions related to Riemann's Zeta function may be calculated.
		We also show how the Stieltjes constants can be evaluated using this scheme and how the Riemann hypothesis can be expressed in terms of the behavior of two of the sequences of coefficients.
		High-precision values for the coefficients of these power series are found using Mathematica.
	library.wolfram.com /infocenter/Articles/788 (88 words)

	Riemann Zeta Function
		is the Euler-Mascheroni constant (Whittaker and Watson 1990, p.
		Here, the sum on the right-hand side is exactly the Dirichlet eta function
		Bailey, D. H.; Borwein, J. M.; and Crandall, R. "On the Khinchin Constant." Math.
	users.skynet.be /fa956617/math/topics/RiemannZetaFunction.html (1820 words)

	PrintThisPage
		A few approximations are given in the following table; present-day technology permits computations of these up to several hundred digits,
		n ranging from 0 to 5 Stieltjes constant gamma(n) 0 0.5772156649 1 -0.07281584548 2 -0.009690363192 3 0.002053834420 4 0.002325370065 5 0.0007933238173
		Briggs, Mitrovic and Bohman & Fröberg have studied the sign patterns of the Stieltjes constants.
	www.mathsoft.com /printThisPage.aspx?1150 (93 words)

	Homepage Carsten Elsner.index.html
		"On a sequence transformation with integral coefficients for Euler’s constant",
		On a sequence transformation with integral coefficients for Euler’s constant, II",
		On sequence transformations with rational coefficients for Euler’s constant and generalized Stieltjes constants"
	www.carstenelsner.de (544 words)

	Texas Section, MAA
		The speaker will generalize this "Lamb product", and discuss his experiences with its use in college mathematics classes taken by preservice teachers of mathematics at various levels.
		Abstract = The Riemann Hypothesis (RH), that no nontrivial zeroes for the Riemann zeta function lie on the line in the complex plane s=1/2, is equivalent to growth and/or positivity behavior of Taylor coefficients in various related complex-analytic functions.
		Our high precision computations of the Stieltjes constants (which are essentially the Laurent series coefficients for zeta centered at s=1) allow us to compute some of these other coefficients to high precision.
	orgs.tamu-commerce.edu /maa/abs-04.html (8131 words)

	Mathematical Constants
		My book Mathematical Constants is now available for online purchase from Cambridge University Press (in the United Kingdom, North America and Australia).
		I mention several favorite links: Mathematical Constants and Computation (by X. Gourdon and P. Sebah), MathWorld Constants (by E. Weisstein at Wolfram Research), and the Inverse Symbolic Calculator and Integer Relations (both at CECM).
		My former employer, MathSoft Inc., has posted my (ancient) draft notes for the book here and here.
	pauillac.inria.fr /algo/bsolve/constant/stltjs/stltjs.html (237 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		What I end up with for the nth coefficient is
		I still can't see where the Stieltjes limit as
		`I still can't see where the Stieltjes limit as`
	mathforum.org /kb/plaintext.jspa?messageID=285248 (136 words)

	Courses and Activities - Analysis
		A fundamental question in the numerical solution of initial value problems for ordinary differential equations is whether the long-time dynamics of ordinary differential equations are preserved under numerical discretization.
		For example one can think of the convergence of solutions to an equilibrium point or a periodic orbit, or of a particular (physical) quantity that remains constant through time.
		In this course, the above question is addressed.
	www.math.leidenuniv.nl /~stieltjes/archief/analysebob.html (1297 words)

	Antibodies to the FVIII light chain that neutralize FVIII procoagulant activity are present in plasma of nonresponder ...
		Calculated association and dissociation constants of the interaction of anti-FVIII IgG with C2 and A2 domains, light chain of FVIII, and intact FVIII
		Determination of kinetic parameters of the binding of affinity-purified anti-FVIII IgG from IVIg to the C2 fragment of FVIII.
		S. Lacroix-Desmazes, J. Bayry, N. Misra, M. Horn, S. Villard, A. Pashov, N. Stieltjes, R. d'Oiron, J.-M. Saint-Remy, J. Hoebeke, M. Kazatchkine, J. Reinbolt, D. Mohanty, and S. Kaveri
	www.bloodjournal.org /cgi/content/full/95/11/3435 (4670 words)

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