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17h
comment Can anyone improve on this work and find a closed form of $\zeta(3)$?
To possibly come nearer an independent description of $\theta$ I'd try and generalize it to the other zetas and thus generate a bunch of thetas and see, whether there is some structure in it...
2d
comment Solve $x^{x^x}=-1$
@Paul: I think myfun(x)=local(lx=(log(x)+6*2*Pi*I));(log(lx)+6*2*Pi*I) + lx*x ; no modification of the mydev() is needed (the branch is handled directly at the call of the logarithm, sorry: I had an error in my previous comment here). But anyway - no guarantee....
Aug
31
comment Solve $x^{x^x}=-1$
@Paul: I think, the other branches are accessible, if in "myfun" and "mydev" the each logarithm-function gets one additional summand of $\log(...) + k \cdot 2 \pi î$ where $k$ is the branchnumbers. (I didn't test this, this is only how I'd start to try)
Aug
31
comment Solve $x^{x^x}=-1$
@renatoFaraone: I don't think we can do it with a finite composition of common /elementary functions. Maybe there is some iterative scheme like it is with the LambertW-implementation, for instance as in wikipedia/rosetta-code... What I'd really like were if one could Fourier-decompose the set of coefficients of $g^{-1}(x)$. Perhaps we get some Fourierseries or Theta-series from this.
Aug
31
comment Solve $x^{x^x}=-1$
@RenatoFaraone: we can have a taylor-series $g(x)=(x+1)^{(x+1)^{(x+1)}}-1$ and then try to evaluate $f(x)=g(x-1)+1$. Because the formal powerseries of $g(x)$ is invertible we can also have $f^{-1}(x) = g^{-1}(x-1)+1$. I don't know about the ranges of convergence of all these power series and functions. But since the signs of the coefficients are seemingly roughly alternating, one can possibly extend the range of convergence to a range of summability using Cesaro-,Euler- or Borel-summation.
Aug
29
comment What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$
See my answer at math.stackexchange.com/questions/1413541/…
Aug
25
comment Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?
I recall a comment of Gerry Myerson, saying that no nontrivial variant of the 3x+1-problem has a solution so far.(Unfortunately I didn't find that comment at the moment. Might be in mathoverflow or here in MSE in some questions tagged with "collatz")
Aug
21
comment How to solve special type of Diophantine equation
I get $ y=0 ,x=0 (0 = 0)$, $y=1 , x=0 (0 = 0)$, $y=2 , x=6 (30 = 30)$, $y=3, x=16 (240 = 240)$, $y=30, x=4930 (24299970 = 24299970)$ for $0 \le y \le 10^6$
Aug
17
comment repeated exponents sign
How can it be difficult to use -for instance- a big E in place for the big $\sum$ resp $\Pi $. I think in wikipedia someone has also proposed a big T (alluding to "T"etration) in place of the $\sum$ resp. $\Pi$
Jul
19
comment How to replace addition with multiplication to find the next integer value?
@siméon: hadn't just this been the point/the nerve of the couriosity of the OP?
Jul
18
comment List of divergent series and their summations
what about wikipedia? There is already a certain list. Why not extend that list?
Jul
15
comment Looking for a verification or refutation my attempted proof of why the Collatz conjecture is probably false.
Have you ever tried to verify the conjecture that $g(x)=2^x$? I've just rund some tests and tried for instance $g(20)$ so 20 steps to reach 1 (or : how much numbers can be generated if you start by n=1 and do the inverse collatz-transformation for 20 steps?). For that example I got $g(20)=321$ or $g(20)=322$ (depending on whether we count the cycle at 1 as well) This comprises from $71$ unique odd numbers plus $250$ additional even numbers as even multiples of the odd unique numbers (only cofactored by perfect powers of 2). See my answer-box for a listing of the odd numbers
Jul
12
comment What is the direction along the edge of a circle called (in English and by chance German)?
@Grantwalzer: Well, now I hope it comes out well for your problem and the ideas were helpful in this or that way.
Jul
11
comment What is the direction along the edge of a circle called (in English and by chance German)?
@Grantwalzer: Also "versetzt" is a better term than "verschieben" in this case - because "verschieben" is somehow felt to be a continuous motion while "versetzt" emphasizes the discreteness of the motion. However - why the limitation on "wachsend" (growing). For a general sentence is "verändert" better, because it doesn't imply only one direction of change of radius. Why should the problem for the reader be specialized to that case of "growing"/"expanding"?
Jul
11
comment What is the direction along the edge of a circle called (in English and by chance German)?
Perhaps "Veränderungen des Radius führen zu Verschiebungen der Punkte auf dem Rand (die Verschiebungen sind nicht-stetig)" . This has also a causal notion. To remove this causal notion I'd say "Veränderungen des Radius gehen einher mit Verschiebungen der Punkte auf dem Rand (die Verschiebungen sind nicht-stetig)". The term "Rand" is here a short form for "auf dem Umfang" ("on the circumference"), and possibly is a bit outdated/today too little technical.
Jul
8
comment A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
I have a Pari/GP-routine which makes it really simple. It uses sumalt , a procedure for divergent summation (based on Zagier's ideas) on formal powerseries. Possibly in Math'ica this is possible as well. I can post it here if you're interested. (Or search for "stieltjes" and my user name, possible I've already posted that routine here or in MO)
Jul
8
comment A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
I assume, the $\gamma$ are the Stieltjes-constants? (I just tried to confirm it numerically using $n=100$,$n=1000$, $n=10000$ and $n=200000$ and seems to approximate Stieltjes $\gamma_1$ and $\gamma_2$
Jul
4
comment Algorithm for tetration to work with floating point numbers
Perhaps this (go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf) is an interesting essay, although it is extremely amateurish (one of my earlier tries to put numbertheoretic puzzles into shape - I had even difficulties to refer to functions having powerseries in the usual way) - I should rewrite this. But it might give a good impression for a first read.
Jul
4
comment Algorithm for tetration to work with floating point numbers
Hmm, I'm not much active on this this days. Look at my oldest postings, I've called that "matrixmethod" (this is perhaps the most fruitful searchable term). And in the beginning I did not know that this all was already known with the term "Carlemanmatrix". There is also some literature on this, look for Aldrovando (the link to the ArXiv by D. Geisler), Eri Jabotinsky, S.C.Woon (perhaps I can come back to this later with more information - we have it extremely hot today which makes it uncomfortable to sort out links...)
Jul
1
comment Super root function
If something useful should exist, it's rather the (analoguous) pentation-"squareroot". Unfortunately the pentation is not remotely so far developed as the tetration. Maybe you find something in the "tetration-forum" math.eretrandre.org/tetrationforum