1d

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Is there any pattern to the continued fraction of $\sqrt[3]{2}$?
For the representation of a continued fraction there is the method using products of 2x2matrices  a set of matrices $M_k$ having the k'th cfcoefficients in its edge; for periodic continued fractions this runs into an eigenvalue problem of the partial product of $P_n =M_1 \times ... \times M_n$ where the matrices contain the periodic tail of the coefficients of the cf. For cubic roots the same can be expressed with 3x3 matrices of the obvious form; and because 3x3matrices of integer entries can have cubic roots in their eigenvalues, that generalization provides then a periodic pattern. 
Nov
7 
answered  How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root 
Oct
28 
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Proofs without words
It needed quite a long time for me to understand this. But, well, then it is amazing! 
Oct
26 
revised 
How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
added image and more precision on the trailing speculation 
Oct
26 
revised 
How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
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Oct
19 
revised 
How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
inserted the term "isogones" and "isogonal" 
Oct
19 
comment 
How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
The hint to the Taylor series was a key which helps me to proceed on my own now and I think I can give and understand answers to Q1 and Q2 now with that means; thank you very much so far 
Oct
19 
accepted  How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root 
Oct
18 
comment 
How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
Upps, maybe there was some misunderstanding. Testing the coefficients of $f(x) = \zeta(x+\rho_0)$ up to index 127 suggests that (with alternating signs) we have summands with nearly constant absolute values when we evaluate $f(\rho_0)$, which is conform to your remark (even more than my first hypothesis of entireness based on 64 terms of the series). What I was thinking about in my previous comment was the range of convergence of the inverse of $f(x)$ which looked like a mercator series by the first 64 terms. 
Oct
18 
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How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
Yes, the original mercatorseries has convergence radius 1, so for any series similar to it the same should be valid (at most). The inverse of the zeta around $\rho_0$ begins as $ \small (1.245 + î0.1982)x + (0.6529  î0.07591)x^2 + (0.4439 + î0.04241)x^3 $ + $ \small (0.3367  î0.02784)x^4 + (0.2714 + î0.02001)x^5 + (0.2274  î0.01524)x^6 $ + $ \small (0.1958 + î0.01209)x^7 + O(x^8)$ and multiplied by $\ln(2)$ the absolute values of the reciprocals of the coefficients are about $\small 0, 1.144, 2.195, 3.235, 4.270, 5.301, 6.330, 7.356,...$ 
Oct
18 
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How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
Yes, that sounds very reasonable. I made the taylorseries $f(x)$ around the first complex root $\rho_0$ and also its seriesinverse $g(x)$; the latter looks very much like a mercatorseries (with a very slight rotation) which suggests a range of convergence of about 1 . I can even compute for instance $ y=g(x\cdot(1+I)) + \rho_0 $ for $x>1$ using Eulersummation (but resulting in a signchange) to arrive at $\zeta(y)=x \cdot (1+I)$). I don't know yet, but it looks very similar that $f(x)$ is entire, at least by the first 64 coefficients of the series. But that cannot really be?? 
Oct
18 
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How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
Checking the same with some other nontrivial roots at the center it might be better to assume some ellipsoid around the root instead of a circle and then to argue from there. 
Oct
18 
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How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
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Oct
18 
revised 
How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
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Oct
18 
asked  How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root 
Oct
9 
comment 
Generating a series representation for the inverse of the operator $f(f)$
Possibly the work of Eri Jabotinski on Carlemanmatrices is relevant here. Carlemanmatrices are such operators, and "halfaoperator" is just the squareroot of the matrix. I don't have all his articles but I think he was the most advanced in this. 
Oct
7 
revised 
Summating divergent series arising from the application of the EulerMaclaurin formula to power law functions with noninteger exponents
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Oct
6 
revised 
Summating divergent series arising from the application of the EulerMaclaurin formula to power law functions with noninteger exponents
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Oct
6 
revised 
Summating divergent series arising from the application of the EulerMaclaurin formula to power law functions with noninteger exponents
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Oct
6 
revised 
Summating divergent series arising from the application of the EulerMaclaurin formula to power law functions with noninteger exponents
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