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Jan
8
comment How does one motivate the analytic continuation of the Riemann zeta function?
Hmm, the question as well this answer is pretty old. It's worth to be noted here anyway, that the question asks for the motivation for the consideration of the "functional equation" which concerns the relation of zeta at negative and positive arguments, not for the motivation to consider the relation between the alternating to the non-alternating series.
Jan
2
comment Why are the only numbers $m$ for which $n^{m+1}\equiv n \bmod m$ also the only numbers such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$?
(at the moment) \pmod is nicely been displayed at my system (Win 7, Firefox)
Dec
14
comment Some Questions on the Collatz conjecture
@Arno: doesn't that idea only say,that ... if we try all numbers to check their equivalence-class, then ... But how can that be seen as an improvement over the original Collatz-transformation (with the (infinite) task of checking each number separately)?
Dec
10
comment Encyclopedia of properties of nonnegative matrices
As I understand this the "totally positive matrix" are a subset of the OP's nonnegative matrices. So I think the qualification in this answer "...of a more general subject,..." is a bit misleading.
Nov
25
comment 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 2x2-matrices - a set of matrices $M_k$ having the k'th cf-coefficients 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 3x3-matrices 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
comment 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
added 885 characters in body
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
comment How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
Yes, the original mercator-series 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
comment 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 series-inverse $g(x)$; the latter looks very much like a mercator-series (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 Euler-summation (but resulting in a sign-change) 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
comment 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
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
revised How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root
added 288 characters in body
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 Carleman-matrices is relevant here. Carleman-matrices are such operators, and "half-a-operator" is just the square-root of the matrix. I don't have all his articles but I think he was the most advanced in this.