f(x,u)=0∫uIm(x1+It(1+It))csch(πt)t=CMRB
f(x,u)=Im()csch(πt)t=CMRB
0
∫
u
x
1+It
(1+It)
Besides x=1, u=∞ what parameters in this form can give the same sum?
What parameters for u and x in the form Integrate[
Csch[Pi t] Im[((1 + I t)^(x/(1 + I t)))], {t, 0, u}] makes it equal the MRB constant?
Csch[Pi t] Im[((1 + I t)^(x/(1 + I t)))], {t, 0, u}] makes it equal the MRB constant?
What parameters in that form that can give MRB constant sum?
m=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision100]
In[]:=
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278
Out[]=
Quiet[N[x]/.FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{x,2},WorkingPrecision100]]
In[]:=
1.
Out[]=
Quiet[x/.FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{x,25},WorkingPrecision100]]
In[]:=
25.65665403510586285599072933607445153794770546058072048626118194900973217186212880099440071247392077
Out[]=
x=25.6566540351058628559907293360744515379477054605807204862611819490097321718621288009944007124739159792146480733342667`100.
In[]:=
25.65665403510586285599072933607445153794770546058072048626118194900973217186212880099440071247391598
Out[]=
m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"]
In[]:=
-9.3470×
-94
10
Out[]=
(results=Table[Quiet[FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,u},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{u,a},WorkingPrecision100]],{a,-3,4}])//TableForm
In[]:=
u-3.205281240093347156628042917392342206845495258334899739232149151500845258217206440409918401888147990 |
u-1.975955817063408761652299553542124207955844914033641475935228264550492865978094627264109660184110813 |
u-1.028853359952178482391753039155168552490590931630510790413956782402957713372878008212396966920395894 |
u0.02332059641642379960870201829771316178976827597634872153712935185694600973532346147587251063375909279 |
u1.028851065679287940491239061962065335882572934488973325479147532561027453451485653303655759243419291 |
u1.975930036556044011032057974393627818460615746707424999683678154612112980778643821981630632223625260 |
u3.377688794565491686010340135121834601964847740798723565007400061225550945863584207850080547919162076 |
u4.218664066279720330451890569753298644162048163522884025276106357736553673659421134883269758395781650 |
Out[]//TableForm=
Quiet[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,u},WorkingPrecision->100,AccuracyGoal->100,Method->"Trapezoidal"]/.results]//TableForm
In[]:=
-1.54387828979211679160225856368628698239013559× -54 10 |
-1.650587080052960541099285654572141729212307× -56 10 |
-4.74733569309890585575136369263660582070431× -57 10 |
3.88432825091562193598182092013732457942482× -57 10 |
4.72926388624601586314520015635582296283049× -57 10 |
1.653990003627211153185296844342954430897517× -56 10 |
3.387517822661408436528755032035501466434285285× -53 10 |
-1.876525309243100801023411326326237599736649923× -53 10 |
Out[]//TableForm=
m-NIntegrate[Csch[Pit]Im[((1+It)^(1/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"]
In[]:=
-9.3472×
-94
10
Out[]=
(results1=Table[Quiet[FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(1/(1+It)))],{t,0,u},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{u,a},WorkingPrecision100]],{a,1,2}])//TableForm
In[]:=
u1.333754341654332447320456098329979657122884399228753780402849117971679458574998581211071632180120751 |
u2.451894470180356539050514838856255986670981536525622143367559422740744344429034215340055689334506038 |
Out[]//TableForm=
Quiet[m-NIntegrate[Csch[Pit]Im[((1+It)^(1/(1+It)))],{t,0,u},WorkingPrecision->100,AccuracyGoal->100,Method->"Trapezoidal"]/.results1]//TableForm
In[]:=
9.14168773509281084175606864178330771020516× -57 10 |
1.29094735553632970875478886497194433642706252× -54 10 |
Out[]//TableForm=