Since M = the MRB constant = sum((-1)^m*(m^(1/m)-1), m = 1 .. infinity)

Maple shows that sum((-1)^m*(m^(1/m)-a), m = 1 .. infinity) = 0 for a = MRB2 = 1- 2*MRB constant.

The Maple input,

evalf(sum((-1)^n*(n^(1/n)-1+2*evalf(sum((-1)^m*(m^(1/m)-1), m = 1 .. infinity), 20)), n = 1 .. infinity), 20);

gives approximately 0 and the larger the precision, (Here it is 20 digits.), the closer the sum is to zero. When the precision is infinate, zero minus the sum, (Which sum does exist.), equals zero. Thus the sum is, at that time, zero.

So I have achieved my two goals which were: find out what value the two limit points of sum((-1)^m*(m^(1/m)-a), m = 1 .. infinity) for a=MRB2 were and prove that MRB2 = 1-2M.

Let a = MRB2 = 1-1-2M = 0.6242807150758..., where M is the MRB constant and the two limit-points of the series f(a) = Sum[(-1)^j*(j^(1/j) - a), {j, Infinity}] converge to +/- MRB constant with its Levin's u-transform's result being 0.

For a not = 1,  series of the form f(a) = Sum[(-1)^n*(n^(1/n)-a),{n,Infinity}] are divergent . I wanted to set the set f(a) = to 0, solve for a and call that value for a "MRB2." I did some bisection method and then resorted to solvers. Using Levin-Type Sequence Transformations Maple and Mathematica were willing to give me a zero, not withstanding, with differing results beyond machine precision even with the use of methods and options. The two results were suspiciously close to MRB2 = 1-1-2M = 0.6242807150758... where M is the MRB constant( that is Sum[(-1)^n*(n^(1/n)-1),{n,Infinity}] ). Now I would like to do two things: Find out precisely where the two limit-points of the series f(a) are converging to when a = MRB2 ( It seems to be +/- MRB constant. ) and find a non trivial proof that shows that MRB2 = 1-2M.

Thank you acer.

It would sure help if I can figure out what the Levin's u-transform's result given through Maple says about the divergent series. I've only started to look up the meaning of it. If anyone can explain it, that would be great!

Through experimentation it seems that MRB2=a=0.62... where

limit(sum((-1)^j*(j^(1/j)-a), j = 1 .. N-1)-(sum((-1)^j*(j^(1/j)-a), j = 1 .. N)), N = infinity)  = 0.376... approximately = 2* MRB*constant

Well I worked with other irrational constants to replace m and found out there exists other pairs of numbers that give similar harmonic qualities to to function f(n) = sin((a+b*n)*pi/M).   But it seems the b has to be different for every constant M. For example, I can use an b as small as 2721 for M=exp(1).


restart; Digits := 52; Seed := randomize(); m := evalf(exp(1)); for a to 10 do u := 10^30*RandomTools[Generate](float); plot(abs(sin((u*m+2721*floor(n))*Pi/m)), n = -1000 .. 1000) end do


Seems to give similar results to what I saw with m= MRB constant. Now instead of focusing on the MRB constant with the function f(n) = sin((a+b*n)*pi/M), I should focus on that function and figure out how I can predict the necessary b for each given constant.


I believe that I will start a new blog about f(n) = sin((a+b*n)*pi/M).


 

Simple trigonometry shows the following:

Let M be any constant not only the MRB constant. With f(n)=sin((a+b*floor(n))*Pi/M), as long as ((a/M)) is close to a multiple of 1/2 f(n) will be near one of the integers {-1, 0, 1}.

Additionally, when b*floor(n)/M is also a multiple of 1/2 all f(n), with reasonably sized n,  will still be near one of the integers {-1, 0, 1}.

I could post some examples but it is easy to see; you might have some fun trying it out yourself. It does takes some time, however, to find some ((a/M)) and b*floor(n)/M that are close enough to a multiple of 1/2 to make the result interesting.

Parlor trick or math? You decide; I did!

The following describes a certain quality of the MRB constant that doesn't appear to be shared with other constants.

Below, m Is the MRB constant.

Let u be any value and multiply  u by the MRB constant add 876799*any integer n, say abs(n)<10^3; divide all that by the MRB constant and multiply by Pi. Take the sine of all of that and the results will always have pretty much a consistent magnitude. The result could have been anywhere within a range of one, but now it will be very close to the same for  all n.

 restart; Digits := 52;Seed:=randomize();

m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity));

for a from 1 to 10 do    u := 10^30*RandomTools[Generate](float);  plot(    abs(   sin((u*m+876799*floor(n))*Pi/m)         ),

              n = -1000 .. 1000); od;
 

Change the value of m even just a little bit and you get results with magnitudes anywhere within the range of abs(sin) for  all n.

 restart; Digits := 52;Seed:=randomize();

m := evalf(0.1+sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity));

for a from 1 to 10 do
                                     u := 10^30*RandomTools[Generate](float);  plot(  abs(    sin((u*m+876799*floor(n))*Pi/m)        ),   n = -1000 .. 1000); od;

There is a much smaller number than the 876799 that produces similar harmonics; it is 1665.

So you can can try the same code using 1655 and get results that are proportional in producing results that have consistent magnitude.

Below we let 10*7*RandomTools[Generate](float) represnt the first few digits of some irrational number.

Compare the  graphs produced by the first few digits of some irrational number used for m in the following,

restart; Digits := 40; m := 10^7*RandomTools[Generate](float); for p to 6 do plot(evalf(1-sin(Pi*(5060936308+78389363*floor(n))/m)), n = 10^p .. 10^(p+1)) end do;


  with the graphs, of the MRB constant used for m, that follow.

restart; Digits := 40; m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); for p to 6 do plot(evalf(1-sin(Pi*(5060936308+78389363*floor(n))/m)), n = 10^p .. 10^(p+1)) end do;

The MRB constant seems to be the unique irrational number that smoothes the graph!
 

Now, if this quality is not unique to the MRB constant then there exists a m for every sin((a+b*n)*Pi/m) that smoothes the graph with respect to n; a and b are real. But I still bet this quality is unique to the MRB constant.

Here are a few plots of the series mentioned above.  Increasing and decreasing the value of ?Digits will drastically alter the appearance of the graphs; for example enter the following:

restart; for Digits from 17 to 20 do m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); plot(evalf(1-sin(Pi*(5060936308+78389363*floor(n))/m)), n = 1 .. 100) end do;

restart; for Digits from 19 to 21 do m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); plot(evalf(1-sin((996509389*2)*floor(n)*Pi/m)), n = 1 .. 500) end do;

restart; for Digits from 20 to 24 do m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); plot(evalf(sin((8737225085429350+1753598*floor(n))*Pi/m)), n = -100 .. 100, discont = true) end do

View 565_set_plots.mw on MapleNet or Download 565_set_plots.mw
View file details

Download near1.mw

WIth many of the above sets of the form sin((a+b*n)*pi/m) centered around ~ -1, you can exchange some of the b and still get ~ -1.

I don’t know if you can do the same thing with other constants or not.  I could propose that you could; as in the following:

There exists large independent sets of approximations -- of the form sin((a+b*n)*pi/m), involving every constant, that produce near integers or near roots of integers -- at periods with integer values. Again, I could say all constants have this quality!

However, after all the thousand of years and combined work of mankind no one has published widely such a quality with Pi or any other constant.  So it is my "best bet" that this quality is unique to the MRB constant or others that are closely related to it like 1-MRB constant.  I am not considering using a=0 with b=0 in my bet.

10121872616/53879973811-m~=0.000000000000000000001 is not significant.  It is just a rational approximation that equals the best accuracy that I've found in large independent sets of approximations -- of the form sin((a+b*n)*pi/m) involving the MRB constant that produce near integers or near roots -- at periods with integer values.  Yes, i do realize that the periods of the sinusoidal functions involved do not have integer values but are very small and it is likely that an occasional computation of the above form will approximate an integer or root with an accuracy of floor[log(10](a/(2Pi))).  However, for even better approximations of predictable values to come at integer intervals is all the more remarkable.  It is my desire to use the existence of these approximations to find an underlying closed form for the MRB constant.