2
cot

β
2

4
π
∫
-π
dω
2π
ωqr
e
N-2
cos
ω
2
sin
ω
(*expandthebinomialinCos[ω]^Landintegratetermbytermanalytically​​pickthes-term(E^(Iωs)E^(-Iω(L-s)))Binomial[L,s]​​*)
In[]:=
HH=Expand[E^(Iωq)TrigToExp[
-2+N
Cos[ω]
Sin[ω]^2]/.(
-ω

+
ω

)^L_:>Binomial[L,s]E^(Iωs-Iω(L-s))]
Out[]=
1-N
2
qω-(-2+N-s)ω+sω

Binomial[-2+N,s]-
-N
2
-2ω+qω-(-2+N-s)ω+sω

Binomial[-2+N,s]-
-N
2
2ω+qω-(-2+N-s)ω+sω

Binomial[-2+N,s]
In[]:=
ClearAll[subInt];
In[]:=
subInt[s_]={E^(X_)F_:>(F/.Solve[D[X,ω]==0,s][[1]])}
Out[]=

X_

F_(F/.Solve[
∂
ω
X0,s]〚1〛)
(*​​thissubstitutionisequivalenttointegrationoverω--selectionoftheconstantterminbinomialexpansion​​*)
In[]:=
JJ[q_]=FullSimplify[HH/.subInt[s]]
Out[]=
-
-N
2
Binomial-2+N,
1
2
(-4+N-q)-2Binomial-2+N,
1
2
(-2+N-q)+Binomial-2+N,
N-q
2

​
In[]:=
1/4Cot[Pip/q]^2Binomial-2+N,
1
2
(-2+N-q)FullSimplifyJJ[q]Binomial-2+N,
1
2
(-2+N-q)/.q->qr
Out[]=
-N
2
(N-
2
q
2
r
)Binomial-2+N,
1
2
(-2+N-qr)
2
Cot
pπ
q

2
N
-
2
q
2
r
(N-
2
q
2
r
)
2
cot

β
2

N
2
(
2
N
-
2
q
2
r
)
N-2
1
2
(N+qr-2)
In[]:=
-N
2
(N-
2
q
2
r
)Binomial-2+N,
1
2
(-2+N-qr)
2
Cot
pπ
q

2
N
-
2
q
2
r
(N(N-1)/2)/4//FullSimplify
Out[]=
-5-N
2
(N-
2
q
2
r
)
2
Cot
pπ
q

Gamma[1+N]
Gamma
1
2
(2+N-qr)Gamma
1
2
(2+N+qr)
-N-5
2
2
cot

πp
q
Γ(N+1)(N-
2
q
2
r
)
Γ
1
2
(N-qr+2)Γ
1
2
(N+qr+2)
In[]:=
asymp=NormalSeries
-5-N
2
(N-N
2
z
)Gamma[1+N]
Gamma
1
2
(2+N-zSqrt[N])Gamma
1
2
(2+N+zSqrt[N])
,{N,Infinity,2}
Out[]=
-
-
2
z
2

N
(-1+
2
z
)
16
2π
+
-
2
z
2

(-1+
2
z
)(3-6
2
z
+
4
z
)
192
N
2π
-
-
2
z
2

(-45+705
2
z
-1230
4
z
+678
6
z
-113
8
z
+5
10
z
)
23040
3/2
N
2π
In[]:=
Integrate[asymp,{z,-Infinity,Infinity}]
Out[]=
0
In[]:=
N
64
π
1/32Zeta[2]/(3Zeta[3])
Out[]=
N
3/2
π
1728Zeta[3]
In[]:=
IIodd[M_?OddQ,q_?OddQ]:=2Sum[
-5-M
2
(M-s^2)Binomial[M,(M+s)/2],{s,q,M,2q}]//Quiet;
In[]:=
testprime=With[{MM=101},ParallelTable[{q,IIodd[MM,q]},{q,3,MM-1,2}]];
In[]:=
Max[N[testprime]]
Out[]=
99.
In[]:=
Min[N[testprime]]
Out[]=
-0.224218
In[]:=
ListPlot[testprime,PlotLabel->"Sum[terms[r!=0]],odd",​​AxesLabel->{N,II}]
Out[]=
In[]:=
IIeven[M_,q_?EvenQ]:=2Sum[
-5-M
2
(M-s^2)Binomial[M,(M+s)/2],{s,q,M,q}]//Quiet;
In[]:=
IIeven[M_,q_?OddQ]:=2Sum[
-5-M
2
(M-s^2)Binomial[M,(M+s)/2],{s,2q,M,2q}]//Quiet;
In[]:=
NormalSeries
-5-N
2
(N-qr^2)Gamma[1+N]
Gamma
1
2
(2+N-qr)Gamma
1
2
(2+N+qr)
,{N,Infinity,2}
Out[]=
N
16
2π
-
1+6
2
qr
64
N
2π
+
1+28
2
qr
+20
4
qr
512
3/2
N
2π
In[]:=
test1=ParallelTable[{M,II[M,M/2]},{M,100000,150000}];
In[]:=
ListPlot[test1]
Out[]=
110000
120000
130000
140000
150000
8.0
8.5
9.0
9.5
In[]:=
lmm=LinearModelFit[test1,{x,x^2},x]
In[]:=
ShowFitLogModel[data_,model_,Ylabel_,title_]:=​​Show[​​ListPlot[data,PlotStyle->{Red,PointSize[0.01]},PlotLegends->{"data"}],​​Plot[Exp[model[Log[x]]],{x,data[[1,1]],data[[-1,1]]},PlotStyle->{Green,Thick},​​PlotLegends->{N[Exp[model[Log[N]]],7]},PlotRange->All],​​PlotLabel->title,​​AxesLabel->{"N",Ylabel}​​];
In[]:=
ShowFitLogModel[test1,loglmm,"Sum[B]",{"BinomialSum[r]"}]
Out[]=
data
0.02493325
0.5000020
N
In[]:=
lognlmm=NonlinearModelFit[Log[test1],{a+1/2x},{a},x]
Out[]=
FittedModel
-3.6915292827711146537+1

In[]:=
Normal[lognlmm]
Out[]=
-3.6915292827711146537+
x
2
In[]:=
ShowFitLogModel[test1,lognlmm,"Sum[B]","BinomialSum[r]"]
Out[]=
data
0.02493384
N