(2E^(Iω(N-2-2k+q))-E^(Iω(N-2k+q))-E^(Iω(N-4-2k+q)))​​2^(-N)(Binomial[N-2,(N+q)/2-1]2-Binomial[N-2,(N+q)/2]-Binomial[N-2,(N+q)/2-2])
In[]:=
2^(-N)Binomial[N-2,(N+q)/2-1]FullSimplify[(2-(Binomial[N-2,(N+q)/2]+Binomial[N-2,(N+q)/2-2])/Binomial[N-2,(N+q)/2-1])]
Out[]=
2-N
2
(N-
2
q
)Binomial-2+N,-1+
N+q
2

2
N
-
2
q
In[]:=
2-N
2
(N-
2
q
)BinomialN-2,
N+q
2
-1
2
N
-
2
q
In[]:=
2-N
2
(N-
2
q
)
(
2
N
-
2
q
)NBeta[(N+q)/2+1,(N-q)/2+1]
FullSimplifyBinomial-2+N,-1+
N+q
2
NBeta[(N+q)/2+1,(N-q)/2+1]
In[]:=
FullSimplify
-N
2
(N-q)(N+q)(N-
2
q
)
N(-1+
2
N
)(
2
N
-
2
q
)
Beta1+
N+q
2
,1+
N-q
2

Out[]=
-N
2
(N-
2
q
)
N(-1+
2
N
)Beta1+
N+q
2
,1+
N-q
2

In[]:=
h[v_,N_]:=Log[1-v^2]-(3+N)Log[2]-LogBeta1+
N+vSqrt[N]
2
,1+
N-vSqrt[N]
2

In[]:=
FullSimplify[Exp[Series[FullSimplify[Normal[Series[h[v,N],{N,Infinity,2}]],{N>1,0<v<1}],{N,Infinity,2}]],{N>1,0<v<1}]
Out[]=
-

-
2
v
2

(-1+
2
v
)
N
4
2π
+
-
2
v
2

(9-3
2
v
-7
4
v
+
6
v
)
1
N
48
2π
-

-
2
v
2

(-1+
2
v
)(-315+60
2
v
+450
4
v
-108
6
v
+5
8
v
)
3/2

1
N

5760
2π
+
2
O
1
N

In[]:=
UniqueFractions[L_]:=Select[Sort[Flatten[UniqueElements[TensorProduct[Range[L],1/Range[L]]]]],#<1&];
In[]:=
UniqueFractions[100];
ProbLog[L_,digits_]:=​​Block{fractions,Tau,taus,LW},​​fractions=Select[Sort[Flatten[UniqueElements[TensorProduct[Range[L],1/Range[L]]]]],#<1&];​​Tau[f_]:=With{M=(L+1)^2,q=Denominator[f],p=Numerator[f]},​​NLog
-3-M
2
(M-
2
q
)
2
Cot
pπ
q

(1+M)Beta1+
M+q
2
,1+
M-q
2

,digits;​​taus=Sort[ParallelTable[Tau[fractions[[i]]],{i,Length[fractions]}]];​​LW=N[Log[1.-(#-0.5)/Length[taus]],16]&/@Range[Length[taus]];​​Transpose[{taus,LW}]​​
fitdata=ProbLog[500,16];
lowdata=Select[fitdata,#[[1]]<1&];​​lowmodel=LinearModelFit[lowdata,{x,x^2,x^3},x]
Out[]=
FittedModel
-0.30272-0.135185x-0.0213599
2
x
-0.0011033
3
x

In[]:=
highdata=Select[fitdata,#[[1]]>2.5&];​​highmodel=LinearModelFit[highdata,{x},x]
Out[]=
FittedModel
0.500467-0.503318x
