the logarithm of the dottie number and Dirichlet’s lambda function.

The Dottie number is the unique real fixed point of the cosine function. (there is a blog about it.) I’m going to use D to denote it. (unfortunately wordpress’s \LaTeX installation isn’t flexible enough to let me use Arakelian’s ա)

D=\cos(D)
therefore
D=\prod_{n=1}^{\infty} \left(1-\frac{4D^{2}}{\pi^{2}(2n-1)^{2}}\right)
therefore
\log D = \sum_{n=1}^{\infty} \log \left(1-\frac{4D^{2}}{\pi^{2}(2n-1)^{2}}\right)
therefore:
\log D = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m} \left[-\frac{4D^{2}}{\pi^{2}(2n-1)^{2}}\right]^{m}
therefore (getting our -1s in order):
\log D = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{2m+1}}{m} \left[\frac{4D^{2}}{\pi^{2}(2n-1)^{2}}\right]^{m}

\log D = \sum_{m=1}^{\infty} \frac{(-1)^{2m+1}}{m} \left(\frac{2D}{\pi}\right)^{2m} \sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2m}}

The last sum is just the Dirichlet Lambda function, so finally, we’ve got:
\log D = \sum_{m=1}^{\infty} \frac{(-1)^{2m+1}}{m} \left(\frac{2D}{\pi}\right)^{2m}\lambda(2m)

(additional: it may be worth investigating the complex fixed points of the cosine function, as far as I know they’re not as well categorized, but the cosine function is entire, so the infinite product identity above should work for the entirety of the complex plane. I may see if the logarithm identity works for any of the complex fixed points)

>>> D = D - (cos(D)-D)/(-1*sin(D)-1)
>>> D
mpc(real='5.8695603773476144', imag='2.5448857668857094')
>>> log(D)
mpc(real='1.8559094433156327', imag='0.40910999959924926')
>>> nsum(lambda n: ((-1)**(2*n+1))*((2*D/pi)**(2*n))*(1/n)*(zeta(2*n)+altzeta(2*n))/2, [1,inf])
mpc(real='1.8559094483018455', imag='-5.8740753096198013')
>>> D
mpc(real='5.8695603773476144', imag='2.5448857668857094')
>>> cos(D)
mpc(real='5.8695603773476153', imag='2.5448857668857099')
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2 responses to “the logarithm of the dottie number and Dirichlet’s lambda function.

  1. Mr. Owen Maresh,

    I wonder why, after a long moderation, my message about the number 0.739085… on your blog http://deoxygerbe.wordpress.com/2011/02/11/the-logarithm-of-the-dottie-number-and-dirichlets-lambda-function/ has been removed. The information introduced there is completely reliable, known to many interested and competent people and none of them disputed it. I hope that this time you will not remove the following, slightly updated message, as I don’t see serious obstacles for presenting it.

    Hrant Arakelian

    The number 0.73908513 … was known in 19th century as a solution of simple transcendental equation cos x = x. In 1981, a quarter of a century before Kaplan’s article in Math. Mag., it was presented in my book as a fundamental mathematical constant and attractor. This book is available in many libraries and is often quoted. Subsequently I designated it by symbol ա (the first letter of the Armenian alphabet) and it appears almost in all of my scientific publications, from the 80th of the last century to the present day. By the way, the book of 1981, as well as an article of 1995 and the symbol ա, are mentioned in MathWorld (http://mathworld.wolfram.com/DottieNumber.html).
    The constant ա is one of two global attractors in the field of simple elementary functions and one of eight (0, e, , i, 2,  – Euler–Mascheroni constant, W(1) – omega constant, ա) fundamental mathematical constants, obtained in the limits of a formal axiomatic system of universal mathematics. This constant and its properties were thoroughly investigated by me as a mathematical quantity (threefold intersection point of functions сos x, arccos x, x; its relation with other FMC’s; its place in the theory of generalized cosine; the first 100000 decimal places of ա are given in http://www.hrantara.com/NewConstant.pdf, 6400000 places (809 pages) in http://www.hrantara.com/NewConstant2.pdf). And above all, it was used (together with other mathematical constants) in constructing the foundations of physical theory, in solving the “inaccessible” problem of theoretical definition and calculation of some physical constants’ numerical values (see e.g. http://www.hrantara.com/Monograph.Abstract.pdf or get my last, published in late 2010 book LMP Fundamental Theory where, alongside with other subjects, is given the whole history of the constant ա).
    All the interested persons (S.Kaplan, P.Blanchard – prof. of mathematics, the husband of Dottie, H.Waldman – Editorial Manager of MathMag, D.Bressoud – president of the MA of America, E.Weisstein, S.Wolfram, N.Sloane) are fully informed of these facts and there are no objections to my naming and notation of the number. “Paul Blanchard is not interested in defending the name” was in the last Kaplan’s letter to me. Certainly, the naming “Dottie number” in your blog and elsewhere is the result of ignorance and misunderstanding. According to scientific ethics and copyright law, the constant and attractor 0.73908513 … ought to be notated by symbol ա and named as a “constant ա” – the simplest form, or “Arakelian’s constant” – by author’s name, or “cosine superposition constant” – by functional location and definition.

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