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Mathematics Recreation Wolfram Language

Help further pattern in the infinite limit of nth eta derivatives of n

Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 10 years ago

One of the growing known formulas for the MRB constant, here B, is where you use consecutive mth Dirichlet eta derivatives of m. (I hope I said that right.) Below etaMM[m, pr] is a module that rapidly computes the above mentioned Dirichlet eta derivatives of m for m>1 to "pr" decimals. Clear[n]; etaMM[m_, pr_] := Module[{a, d, s, k, b, c}, a[j_] := Log[j + 1]^m/(j + 1)^m; n = Floor[1.32 pr]; d = Cos[n ArcCos[3]]; {b, c, s} = {-1, -d, 0}; Do[c = b - c; s = s + c a[k]; b = (k + n) (k - n) b/((k + 1) (k + 1/2)), {k, 0, n - 1}]; Return[N[s/d, pr] (-1)^m]]; When experimenting with etaMM it seemed that the limit etaMM(x)/etaMM(x+1) as x goes to infinity converged to -3/Log[3] as x grew large. So I entered the following. Limit[etaMM[x, 30]/etaMM[x + 1, 30], x -> Infinity] -2.730717679880512180842720 N[-3/Log[3], 30] -2.73071767988051218084272049721 I then put my attention on the similar limit of etaMM(x)/etaMM(x+2) and entered, Limit[etaMM[x, 30]/etaMM[x + 2, 30], x -> Infinity] 7.4568190472120073993794257 N[(-3/Log[3])^2, 30] 7.45681904721200739937942566098 I then looked to see if -3/Log[3])^n fit the pattern for the above mentioned limit of etaMM(x)/etaMM(x+n) by entering the following command. Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - N[(-3/Log[3])^n, 30], {n, 1, 10}] {-1.3723528794080082216904152448810^40, 2.7944491053542312987951080770410^41, \ -7.6308515775170794410843764373110^41, 2.0837701315269985518423223309510^42, \ -4.2430701805510494125169165411110^43, 1.1586626759004547570908228217110^44, \ -3.1639806540990356489457321554010^44, 6.4426453581087240546965967872510^45, \ -1.7593045584587606494957102422010^46, \ -1.3118815741362196959328895546710^47} Please join in the conversation if you have anything to add! In particular I would like to find a shortcut for finding etaMM(x+n) from etaMM(x). This may be hard to do because etaMM(x) is not at all well behaved for small x and works its way to near 0 for larger x.; see the following. ListPlot[Table[etaMM[1 + x, 30], {x, 1, 10}]] It is my guess that etaMM is a decaying, semi-oscillating function: ListPlot[Table[etaMM[1 + x, 30], {x, 1, 10}], PlotRange -> {{1, 10}, {-0.005, 0.005}}, Joined -> True] Table[etaMM[1 + x, 30], {x, 1, 10}] // TableForm

One of the growing known formulas for the MRB constant, here B, is enter image description here

where you use consecutive mth Dirichlet eta derivatives of m. (I hope I said that right.)

Below etaMM[m, pr] is a module that rapidly computes the above mentioned Dirichlet eta derivatives of m for m>1 to "pr" decimals.

Clear[n]; 
etaMM[m_, pr_] := 
 Module[{a, d, s, k, b, c}, a[j_] := Log[j + 1]^m/(j + 1)^m;
  n = Floor[1.32 pr];
  d = Cos[n ArcCos[3]];
  {b, c, s} = {-1, -d, 0};
  Do[c = b - c;
   s = s + c a[k];
   b = (k + n) (k - n) b/((k + 1) (k + 1/2)), {k, 0, n - 1}];
  Return[N[s/d, pr] (-1)^m]];

When experimenting with etaMM it seemed that the limit etaMM(x)/etaMM(x+1) as x goes to infinity converged to -3/Log[3] as x grew large. So I entered the following.

Limit[etaMM[x, 30]/etaMM[x + 1, 30], x -> Infinity]

-2.730717679880512180842720

N[-3/Log[3], 30]

-2.73071767988051218084272049721

I then put my attention on the similar limit of etaMM(x)/etaMM(x+2) and entered,

Limit[etaMM[x, 30]/etaMM[x + 2, 30], x -> Infinity]

7.4568190472120073993794257

N[(-3/Log[3])^2, 30]

7.45681904721200739937942566098

I then looked to see if -3/Log[3])^n fit the pattern for the above mentioned limit of etaMM(x)/etaMM(x+n) by entering the following command.

Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - 
  N[(-3/Log[3])^n, 30], {n, 1, 10}]

{-1.37235287940800822169041524488*10^40, 
 2.79444910535423129879510807704*10^41, \
-7.63085157751707944108437643731*10^41, 
 2.08377013152699855184232233095*10^42, \
-4.24307018055104941251691654111*10^43, 
 1.15866267590045475709082282171*10^44, \
-3.16398065409903564894573215540*10^44, 
 6.44264535810872405469659678725*10^45, \
-1.75930455845876064949571024220*10^46, \
-1.31188157413621969593288955467*10^47}

Please join in the conversation if you have anything to add! In particular I would like to find a shortcut for finding etaMM(x+n) from etaMM(x). This may be hard to do because etaMM(x) is not at all well behaved for small x and works its way to near 0 for larger x.; see the following.

ListPlot[Table[etaMM[1 + x, 30], {x, 1, 10}]]

enter image description here

It is my guess that etaMM is a decaying, semi-oscillating function:

ListPlot[Table[etaMM[1 + x, 30], {x, 1, 10}], 
 PlotRange -> {{1, 10}, {-0.005, 0.005}}, Joined -> True]

enter image description here

Table[etaMM[1 + x, 30], {x, 1, 10}] // TableForm

enter image description here

POSTED BY: Marvin Ray Burns

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Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 10 years ago

As some of you already figured out, something is awry with my use and interpretation of Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - N[(-3/Log[3])^n, 30], {n, 1, 10}] {-1.3723528794080082216904152448810^40, 2.7944491053542312987951080770410^41, \ -7.6308515775170794410843764373110^41, 2.0837701315269985518423223309510^42, \ -4.2430701805510494125169165411110^43, 1.1586626759004547570908228217110^44, \ -3.1639806540990356489457321554010^44, 6.4426453581087240546965967872510^45, \ -1.7593045584587606494957102422010^46, \ -1.3118815741362196959328895546710^47} because, to the contrary, I can enter and get the following. In[13]:= Limit[etaMM[x, 30 + 2]/etaMM[x + 2, 30 + 2], x -> Infinity] - N[(-3/Log[3])^2, 30] Out[13]= 0.10^-28 In[11]:= Limit[etaMM[x, 30]/etaMM[x + 3, 30], x -> Infinity] - N[(-3/Log[3])^3, 30] Out[11]= 0.10^-26 In[16]:= Limit[etaMM[x, 30]/etaMM[x + 4, 30], x -> Infinity] - N[(-3/Log[3])^4, 30] Out[16]= 0.10^-25 It looks like I've got n doing double work; it's global in etaMM and the table both. This works, though: In[58]:= Table[ Limit[etaMM[x, 30 + u]/etaMM[x + u, 30 + u], x -> Infinity] - N[(-3/Log[3])^u, 30], {u, 1, 10}] Out[58]= {0.10^-26, 0.10^-28, 0.10^-29, 0.10^-29, 0.10^-28, 0.10^-28, 0.10^-27, 0.10^-27, 0.10^-27, 0.*10^-26}

As some of you already figured out, something is awry with my use and interpretation of

Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - 
  N[(-3/Log[3])^n, 30], {n, 1, 10}]

{-1.37235287940800822169041524488*10^40, 
 2.79444910535423129879510807704*10^41, \
-7.63085157751707944108437643731*10^41, 
 2.08377013152699855184232233095*10^42, \
-4.24307018055104941251691654111*10^43, 
 1.15866267590045475709082282171*10^44, \
-3.16398065409903564894573215540*10^44, 
 6.44264535810872405469659678725*10^45, \
-1.75930455845876064949571024220*10^46, \
-1.31188157413621969593288955467*10^47}

because, to the contrary, I can enter and get the following.

In[13]:= Limit[etaMM[x, 30 + 2]/etaMM[x + 2, 30 + 2], x -> Infinity] -
  N[(-3/Log[3])^2, 30]

Out[13]= 0.*10^-28

In[11]:= Limit[etaMM[x, 30]/etaMM[x + 3, 30], x -> Infinity] - 
 N[(-3/Log[3])^3, 30]

Out[11]= 0.*10^-26

In[16]:= Limit[etaMM[x, 30]/etaMM[x + 4, 30], x -> Infinity] - 
 N[(-3/Log[3])^4, 30]

Out[16]= 0.*10^-25

It looks like I've got n doing double work; it's global in etaMM and the table both. This works, though:

In[58]:= Table[
 Limit[etaMM[x, 30 + u]/etaMM[x + u, 30 + u], x -> Infinity] - 
  N[(-3/Log[3])^u, 30], {u, 1, 10}]

Out[58]= {0.*10^-26, 0.*10^-28, 0.*10^-29, 0.*10^-29, 0.*10^-28, 
 0.*10^-28, 0.*10^-27, 0.*10^-27, 0.*10^-27, 0.*10^-26}

POSTED BY: Marvin Ray Burns

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