Riesz s - energy (s<d and s≠0, singular case)

Case d == 2 (0 < s < d )

Kernel expansion and remainder

d=2;
Sum[(2^(-(s/2)))((Pochhammer[(s/2),m])/(m!))(t^m),{m,0,∞},Assumptions->{0 < s < d}]

Clear[d];

d=2;
Sum[(2^(-(s/2)))((Pochhammer[(s/2),m])/(m!))(t^m),{m,M+1,∞},Assumptions->{0 < s < d}]

Clear[d];

Linear transformation Abramowitz and Stegun, 15.3 .3

HHypergeometric2F1[1,1+M+(s/2),2+M,t]/.ABR15x3xx3

(2^(-s/2))((Pochhammer[(s/2),1+M])/((1+M)!)) (((t^(1+M)) )/(((1-t))^(s/2)))Hypergeometric2F1[1-(s/2),1+M,2+M,t] ;

When following is positive, remainder (starting with even index term) can be resumed to a positive sum

((Pochhammer[1-(s/2),k]Pochhammer[1+M,k])/(Pochhammer[2+M,k]k!))//FunctionExpand
((%/.k->k+1)/(%/.k->k))//FullSimplify
Reduce[%<1&&0<s<2&&M≥1&&k≥0]

((Pochhammer[1-(s/2),k]Pochhammer[1+M,k])/(Pochhammer[2+M,k]k!));
(%/.k->k)-(%/.k->k+1)//FunctionExpand//FullSimplify

... positive for 0 < s < 2.

First resumation of leading terms (coefficient (Subscript[A, d])(M))

((1+(((-1))^m))/2)(2^(-(s/2)))((Pochhammer[(s/2),m])/(m!))((Pochhammer[(1/2),(m/2)])/(Pochhammer[((d+1)/2),(m/2)]))/.d->2//FullSimplify

s=.;
TMP=Sum[(((2^(-(s/2))) Gamma[2r+(s/2)])/(Gamma[2+2r] Gamma[(s/2)])),{r,0,((M-1)/2)},Assumptions->{0<s<2}]//Distribute
Clear[s];

TMP==((2^(1-s))/(2-s))-(((2^(-s/2)) Gamma[1+M+(s/2)])/(Gamma[3+M] Gamma[(s/2)]))HypergeometricPFQ[{1,(1/2)+(M/2)+(s/4),1+(M/2)+(s/4)},{(3/2)+(M/2),2+(M/2)},1]//FullSimplify

Wsd=(2^(d-s-1))((Gamma[((d+1)/2)]Gamma[((d-s)/2)])/((Sqrt[π])Gamma[d-(s/2)]))/.d->2//FullSimplify

Application of Prudnikov etal III, 7.4 .4 .1

HHypergeometricPFQ[{(1/2)+(M/2)+(s/4),1+(M/2)+(s/4),1},{(3/2)+(M/2),2+(M/2)},1]/.PBMx7x4x4xx1

(Subscript[A, d])[M]==((2^(1-s))/(2-s))-(((2^(-s/2)) Gamma[1+M+(s/2)])/(Gamma[3+M] Gamma[(s/2)]))((1+M)/(2-s))HypergeometricPFQ[{(3/2)-(s/4),1-(s/4),1},{2-(s/2),2+(M/2)},1];

((2^(1-s))/(2-s))-(((2^(-s/2)) Gamma[1+M+(s/2)])/(Gamma[3+M] Gamma[(s/2)]))((1+M)/(2-s))Sum[((Pochhammer[(3/2)-(s/4),ell]Pochhammer[1-(s/4),ell])/(Pochhammer[2-(s/2),ell]Pochhammer[2+(M/2),ell])),{ell,0,10}];
Series[%,{M,∞,2}]

Sum[(2^(-(s/2)))((Pochhammer[(s/2),m])/(m!)),{m,0,M}]//FullSimplify//Distribute

((2^(1-s))/(2-s))-(((2^(-s/2)) Gamma[1+M+(s/2)])/(Gamma[3+M] Gamma[(s/2)]))((1+M)/(2-s))Sum[((Pochhammer[(3/2)-(s/4),ell]Pochhammer[1-(s/4),ell])/(Pochhammer[2-(s/2),ell]Pochhammer[2+(M/2),ell])),{ell,0,10}]-(1/K)(((2^(-s/2)) Gamma[1+M+(s/2)])/(Gamma[1+M] Gamma[1+(s/2)]));
Series[%(K^2)/.M->c K + δ,{K,∞,1},Assumptions->{c>0,0≤δ<2}]//FullSimplify//Distribute

((2^(1-s))/(2-s))(K^2)+((((2^(-s/2)) (c^(-1+(s/2))) (-4 c-s+2 c s) )/((2-s) s Gamma[(s/2)])))(K^(1+(s/2)));

F=(((2^(-1-s/2)) (c^(-1+(s/2))) (-4 c-s+2 c s) )/((2-s) Gamma[1+(s/2)]));

Plot3D[F,{c,0,2},{s,0,2},AxesLabel->Automatic]

Fp=D[F,c]//FullSimplify

sol=Solve[Fp==0,c]//Flatten

F/.sol//FullSimplify

((2^(1-s))/(2-s))(K^2)-(((2^(1-s))/((2-s)  Gamma[1+(s/2)])))(K^(1+(s/2)))

d=2; (* hexagonal lattice *)

covolume=((Sqrt[3])/2);
ZetaA2=6Zeta[s](3^(-s))(Zeta[s,(1/3)]-Zeta[s,(2/3)]);

ωd=((2(π^((d+1)/2)))/(Gamma[((d+1)/2)]));
Wsd=(2^(d-s-1))((Gamma[((d+1)/2)]Gamma[((d-s)/2)])/((Sqrt[π])Gamma[d-(s/2)]));
Csd=(covolume^(s/d))(ZetaA2/.s->(s/2));

Print["conjectured"];
boundconj=(Csd/((ωd)^(s/d)))//FunctionExpand//FullSimplify
N[%,24];

(* Series[Wsd,{s,d,1}]//FullSimplify

Series[(Csd/((ωd)^(s/d))),{s,d,1}]//FullSimplify*)

Print["known bound"];
Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)]));
ctilde=-(((2^(-(s/2)))(s^(-(s/2))))/(1-(s/2)));
N[ctilde,24];

Print["new bound"];
boundnew=-(((2^(1-s))/((2-s)  Gamma[1+(s/2)])));
N[cstar[d,d],24];

plot=Plot[{boundconj,boundnew,ctilde},{s,0,2},PlotStyle->{Dashed,Automatic,Dotted},WorkingPrecision->128]

Print["comparision (absolute and relative)"];

plotA=Plot[{Abs[((boundconj-boundnew)/boundconj)],boundconj-boundnew},{s,0,2},WorkingPrecision->128,PlotStyle->{Dashed,Automatic,Dotted},GridLines->{None,{0.025}},PlotRange->{All,{0,0.025}}]

plotB=Plot[{Abs[((boundconj-ctilde)/boundconj)],boundconj-ctilde},{s,0,2},WorkingPrecision->128,PlotStyle->{Dashed,Automatic,Dotted},GridLines->{None,{0.35}},PlotRange->{All,{0,0.35}}]



(* Plot[{boundconj,boundnew},{s,0,2},WorkingPrecision->64,Epilog->Inset[plotA,{-3,0.1},{.5,0},1]]*)



Clear[ωd,Wsd,Csd,d,Cd];

Riesz d - energy (boundary case)

AdM[2,M_]:=(1/4) (PolyGamma[0,1+(M/2)]+EulerGamma+2Log[2]);
AdM[4,M_]:=(3/8) (PolyGamma[0,2+(M/2)]+EulerGamma+2Log[2]-2);
AdM[8,M_]:=(35/64)(PolyGamma[0,4+(M/2)]+EulerGamma+2 Log[2]-(11/3)+(3/(6+M)));
AdM[24,M_]:=((2028117 )/2097152)(PolyGamma[0,12+(M/2)]+EulerGamma+2 Log[2]-(83711/13860)+(693/(128 (14+M)))-(7161/(64 (16+M)))+(9471/(16 (18+M)))-(71511/(64 (20+M)))+(87923/(128 (22+M))));

d == 2 (hexagonal lattice)

d=2;

Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)])) (* leading term constant *)

Series[(AdM[d,M]-(2^(-d/2))(( Pochhammer[1+(d/2),M])/(M!))(1/K))(K^2)/.M->(((c K))^(2/d))+δ,{K,∞,-1},Assumptions->{c>0,0≤δ<2}]//FullSimplify[#,Assumptions->{c>0,0≤δ<2}]&

Clear[d];

Leading terms of asymptotics

TMP =(1/4) (-2 c+EulerGamma+Log[2 c K]) (K^2)-((((-1+2 c) (1+δ)) K)/(4 c));
TMP==(1/4)(K^2)Log[K]+(1/4) (EulerGamma+Log[2]+Log[c]-2c) (K^2)-((((-1+2 c) (1+δ)) )/(4 c))K//FullSimplify[#,Assumptions->{c>0,K>0}]&

d=2;

(1/4)==Cd
(1/4)==(d/2)Cd

Clear[d];

Optimizing function of (N^2)-term

F=(1/4) (EulerGamma+Log[2]+Log[c]-2c);

Plot[F,{c,0,2}]

Fp=D[F,c]//FullSimplify

sol=Solve[Fp==0,c]//Flatten

F/.sol//FullSimplify

N[%,24]

cstar[2,2](* cstar[s_,d_] *)=(1/4) (EulerGamma-1);

N[cstar[2,2],24]

-((((-1+2 c) (1+δ)) )/(4 c))K/.sol//FullSimplify

0;

Comparison with known bound and conjecture

d=2; (* hexagonal lattice *)

covolume=((Sqrt[3])/2);
ZetaA2=6Zeta[s](3^(-s))(Zeta[s,(1/3)]-Zeta[s,(2/3)])

ωd=((2(π^((d+1)/2)))/(Gamma[((d+1)/2)]));
Wsd=(2^(d-s-1))((Gamma[((d+1)/2)]Gamma[((d-s)/2)])/((Sqrt[π])Gamma[d-(s/2)]));
Csd=(covolume^(s/d))(ZetaA2/.s->(s/2));

Print["conjectured"];
boundconj=Limit[Wsd+(Csd/((ωd)^(s/d))),s->d]//FunctionExpand//FullSimplify
N[%,24]

(* Series[Wsd,{s,d,1}]//FullSimplify

Series[(Csd/((ωd)^(s/d))),{s,d,1}]//FullSimplify*)

Print["known bound"];
Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)]));
ctilde=-Cd(1-Log[Cd]+d(PolyGamma[(d/2)]+EulerGamma-Log[2]))//FullSimplify
N[ctilde,24]

Print["new bound"];
N[cstar[d,d],24]

Print["comparision (absolute and relative)"];

N[boundconj-cstar[d,d],24]

N[Abs[((boundconj-cstar[d,d])/(cstar[d,d]))100],24]

Clear[ωd,Wsd,Csd,d,Cd];

Cconj[2,2](* Cconj[s_,d_] *)=(1/4)(EulerGamma-Log[2(Sqrt[3]) π])+((Sqrt[3])/(4π))(StieltjesGamma[1,(2/3)]-StieltjesGamma[1,(1/3)]);

N[Cconj[2,2],24]

d == 4 (checkerboard lattice)

d=4;

Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)])) (* leading term constant *)

Series[(AdM[d,M]-(2^(-d/2))(( Pochhammer[1+(d/2),M])/(M!))(1/K))(K^2)/.M->(((c K))^(2/d))+δ,{K,∞,-1},Assumptions->{c>0,0≤δ<2}]//FullSimplify[#,Assumptions->{c>0,0≤δ<2}]&

Clear[d];

Leading terms of asymptotics

TMP =(1/16) (-2 (6+c-3 EulerGamma)+3 Log[4 c K]) (K^2)+(((9-3 c+3 δ-2 c δ) (K^(3/2)))/(8 (Sqrt[c])));
TMP==(3/16)(K^2)Log[K]+(3/8) (EulerGamma+Log[2]-2+((Log[c])/2)-(c/3)) (K^2)+((9-3 c-(-3+2 c) δ )/(8 (Sqrt[c])))  (K^(3/2))//FullSimplify[#,Assumptions->{c>0,K>0}]&

d=4;

(3/16)==Cd
(3/8)==(d/2)Cd

Clear[d];

Optimizing function of (N^2)-term

F=(3/8) (EulerGamma+Log[2]-2+((Log[c])/2)-(c/3));

Plot[F,{c,0,2}]

Fp=D[F,c]//FullSimplify

sol=Solve[Fp==0,c]//Flatten

F/.sol//FullSimplify

N[%,24]

cstar[4,4](* cstar[s_,d_] *)=(3/8)(EulerGamma+(1/2)Log[2]+(1/2)Log[3]-(5/2));

N[cstar[4,4],24]

((9-3 c-(-3+2 c) δ )/(8 (Sqrt[c])))  (K^(3/2))/.sol//FullSimplify

(3/8) (Sqrt[(3/2)]) (K^(3/2));

Comparison with known bound and conjecture

d=4; (* checkerboard lattice *)

covolume=1;
ZetaD4=24 (2^(-(s/2)))(1-(2^(1-s)))Zeta[s]Zeta[s-1]//FullSimplify//FunctionExpand

ωd=((2(π^((d+1)/2)))/(Gamma[((d+1)/2)]));
Wsd=(2^(d-s-1))((Gamma[((d+1)/2)]Gamma[((d-s)/2)])/((Sqrt[π])Gamma[d-(s/2)]));
Csd=(covolume^(s/d))(ZetaD4/.s->(s/2));

Print["conjectured"];
boundconj=Limit[Wsd+(Csd/((ωd)^(s/d))),s->d]
N[%,24]

(* Series[Wsd,{s,d,1}]//FullSimplify

Series[(Csd/((ωd)^(s/d))),{s,d,1}]//FullSimplify*)

Print["known bound"];
Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)]));
ctilde=-Cd(1-Log[Cd]+d(PolyGamma[(d/2)]+EulerGamma-Log[2]))//FullSimplify
N[ctilde,24]

Print["new bound"];
N[cstar[d,d],24]

Print["comparision (absolute and relative)"];

N[boundconj-cstar[d,d],24]

N[Abs[((boundconj-cstar[d,d])/(cstar[d,d]))100],24]

Clear[ωd,Wsd,Csd,d,Cd];

Cconj[4,4](* Cconj[s_,d_] *)=(3/8) (-1+EulerGamma+Log[2]+(1/2)Log[3]+((6 Zeta'[2])/(π^2))-Log[π]);

N[Cconj[4,4],24]

d == 8 ((Subscript[E, 8]) lattice)

d=8;

Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)])) (* leading term constant *)

Series[(AdM[d,M]-(2^(-d/2))(( Pochhammer[1+(d/2),M])/(M!))(1/K))(K^2)/.M->(((c K))^(2/d))+δ,{K,∞,-1},Assumptions->{c>0,0≤δ<2}]//FullSimplify[#,Assumptions->{c>0,0≤δ<2}]&

Clear[d];

Leading terms of asymptotics

TMP = (1/768) (-2 c+35 (-44+12 EulerGamma+Log[4096]+3 Log[c K])) (K^2)+(((105 (10+δ)-c (5+2 δ)) (K^(7/4)))/(192 (c^(1/4))));
TMP==(35/256)(K^2)Log[K]+(35/64) (EulerGamma+Log[2]-(11/3)+(1/4)Log[c]-(c/210)) (K^2)+(((1050-5 c-(-105+2 c) δ))/(192 (c^(1/4))))  (K^(7/4))//FullSimplify[#,Assumptions->{c>0,K>0}]&

d=8;

(35/256)==Cd
(35/64)==(d/2)Cd

Clear[d];

Optimizing function of (N^2)-term

F=(35/64) (EulerGamma+Log[2]-(11/3)+(1/4)Log[c]-(c/210));

Plot[F,{c,0,100}]

Fp=D[F,c]//FullSimplify

sol=Solve[Fp==0,c]//Flatten

F/.sol//FullSimplify

N[%,24]

cstar[8,8](* cstar[s_,d_] *)=(35/64)(EulerGamma+(3/4)Log[2]+(1/4)Log[3]+(1/4)Log[5]+(1/4)Log[7]-(47/12));

N[cstar[8,8],24]

(((1050-5 c-(-105+2 c) δ))/(192 (c^(1/4))))  (K^(7/4))/.sol//FullSimplify

(5/64) ((((105/2)))^(3/4)) (K^(7/4));

Comparison with known bound and conjecture

d=8; (* E8 lattice *)

covolume=1;
ZetaE8=240 (2^(-s))Zeta[s]Zeta[s-3]//FullSimplify//FunctionExpand


ωd=((2(π^((d+1)/2)))/(Gamma[((d+1)/2)]));
Wsd=(2^(d-s-1))((Gamma[((d+1)/2)]Gamma[((d-s)/2)])/((Sqrt[π])Gamma[d-(s/2)]));
Csd=(covolume^(s/d))(ZetaE8/.s->(s/2));

Print["conjectured"];
boundconj=Limit[Wsd+(Csd/((ωd)^(s/d))),s->d]
N[%,24]

(* Series[Wsd,{s,d,1}]//FullSimplify

Series[(Csd/((ωd)^(s/d))),{s,d,1}]//FullSimplify*)

Print["known bound"];
Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)]));
ctilde=-Cd(1-Log[Cd]+d(PolyGamma[(d/2)]+EulerGamma-Log[2]))//FullSimplify
N[ctilde,24]

Print["new bound"];
N[cstar[d,d],24]

Print["comparision (absolute and relative)"];

N[boundconj-cstar[d,d],24]

N[Abs[((boundconj-cstar[d,d])/(cstar[d,d]))100],24]

Clear[ωd,Wsd,Csd,d,Cd];

Cconj[8,8](* Cconj[s_,d_] *)=(35/64) (-(11/6)+EulerGamma-(1/4)Log[2]+(1/4)Log[3]+(1/4)Log[5]+(1/4)Log[7]+((90 )/(π^4))Zeta'[4]-Log[π]);

N[Cconj[8,8],24]

d == 24 (Leech lattice)

d=24;

Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)])) (* leading term constant *)

Series[(AdM[d,M]-(2^(-d/2))(( Pochhammer[1+(d/2),M])/(M!))(1/K))(K^2)/.M->(((c K))^(2/d))+δ,{K,∞,-1},Assumptions->{c>0,0≤δ<2}]//FullSimplify[#,Assumptions->{c>0,0≤δ<2}]&

Clear[d];

Leading terms of asymptotics

TMP = ((-2 c+273795795 (-83711+13860 EulerGamma+13860 Log[2]+1155 Log[c K])) (K^2))/3923981107200+(((316234143225 (78+δ)-c (13+2 δ)) (K^(23/12)))/(326998425600 (c^(1/12))));
TMP==(676039/8388608)(K^2)Log[K]+(2028117/2097152) (EulerGamma+Log[2]+((Log[c])/12)-(c/1897404859350)-(83711/13860)) (K^2)+(((24666263171550-13 c-(-316234143225+2 c) δ))/(326998425600 (c^(1/12)))) (K^(23/12))//FullSimplify[#,Assumptions->{c>0,K>0}]&

d=24;

(676039/8388608)==Cd
(2028117/2097152)==(d/2)Cd

Clear[d];

Optimizing function of (N^2)-term

F=(2028117/2097152) (EulerGamma+Log[2]+((Log[c])/12)-(c/1897404859350)-(83711/13860));

Plot[F,{c,0,316234143225}]

Fp=D[F,c]//FullSimplify

sol=Solve[Fp==0,c]//Flatten

F/.sol//FullSimplify

N[%,24]

cstar[24,24](* cstar[s_,d_] *)=((2028117 )/2097152)(EulerGamma+(11/12)Log[2]+(5/12)Log[3]+(1/6)Log[5]+(1/6)Log[7]+(1/12)Log[11]+(1/12)Log[13]+(1/12)Log[17]+(1/12)Log[19]+(1/12)Log[23]-(42433/6930));

N[cstar[24,24],24]

(((24666263171550-13 c-(-316234143225+2 c) δ))/(326998425600 (c^(1/12)))) (K^(23/12))/.sol//FullSimplify

((13 (3^(7/12)) (7^(5/6)) (1062347^(11/12)) )/(2097152 (2^(11/12)) (5^(1/6))))(K^(23/12));

Comparison with known bound and conjecture

d=24; (* Leech *)

covolume=1;
ZetaΛ24=(65520/691) (2^(-s))(Zeta[s]Zeta[s-11]-RamanujanTauL[s])//FullSimplify//FunctionExpand


ωd=((2(π^((d+1)/2)))/(Gamma[((d+1)/2)]));
Wsd=(2^(d-s-1))((Gamma[((d+1)/2)]Gamma[((d-s)/2)])/((Sqrt[π])Gamma[d-(s/2)]));
Csd=(covolume^(s/d))(ZetaΛ24/.s->(s/2));

Print["conjectured"];
boundconj=Limit[Wsd+(Csd/((ωd)^(s/d))),s->d]
N[%,24]

(* Series[Wsd,{s,d,1}]//FullSimplify

Series[(Csd/((ωd)^(s/d))),{s,d,1}]//FullSimplify*)

Print["known bound"];
Cd=(1/d)((Gamma[((d+1)/2)])/((Sqrt[π])Gamma[(d/2)]));
ctilde=-Cd(1-Log[Cd]+d(PolyGamma[(d/2)]+EulerGamma-Log[2]))//FullSimplify
N[ctilde,24]

Print["new bound"];
N[cstar[24,24],24]

Print["comparision (absolute and relative)"];

N[boundconj-cstar[24,24],24]

N[Abs[((boundconj-cstar[24,24])/(cstar[24,24]))100],24]

Clear[ωd,Wsd,Csd,d,Cd];

Cconj[24,24](* Cconj[s_,d_] *)=(2028117/2097152) (EulerGamma-((Log[2])/12)+((5 Log[3])/12)+((Log[5])/6)+((Log[7])/6)+((Log[11])/12)+((Log[13])/12)+((Log[17])/12)+((Log[19])/12)+((Log[23])/12)-(83711/27720)+((638512875 )/(691 (π^12)))Zeta'[12]-((638512875 )/(691 (π^12)))RamanujanTauL[12]-Log[π]);

N[Cconj[24,24],24]

Proof of CBHS--C22 relation

Numerics

CBHS=2Log[2]+(1/2)Log[(2/3)]+3Log[((Sqrt[π])/(Gamma[(1/3)]))];

C22=(1/4)(EulerGamma-Log[2(Sqrt[3])π])+((Sqrt[3])/(4π))(StieltjesGamma[1,(2/3)]-StieltjesGamma[1,(1/3)]);

CBHS-2C22==Log[2]-EulerGamma//FullSimplify

Block[{$MaxExtraPrecision=1000},N[CBHS-2C22-(Log[2]-EulerGamma),256]]

Malmsten' s identity

m=1;
n=3;

REF=StieltjesGamma[1,(m/n)]-StieltjesGamma[1,1-(m/n)]
RES=2π Sum[Sin[((2π m ell)/n)]Log[Gamma[(ell/n)]],{ell,1,n-1}]-π (EulerGamma+Log[2π n])Cot[((π m)/n)]

N[REF-RES,32]

Block[{$MaxExtraPrecision=1000},N[REF-RES,16]]

Clear[m,n];

m=2;
n=3;

REF=StieltjesGamma[1,(m/n)]-StieltjesGamma[1,1-(m/n)]
RES=2π Sum[Sin[((2π m ell)/n)]Log[Gamma[(ell/n)]],{ell,1,n-1}]-π (EulerGamma+Log[2π n])Cot[((π m)/n)]

CBHS=2Log[2]+(1/2)Log[(2/3)]+3Log[((Sqrt[π])/(Gamma[(1/3)]))];

C22=(1/4)(EulerGamma-Log[2(Sqrt[3])π])+((Sqrt[3])/(4π))(RES (* StieltjesGamma[1,(2/3)]-StieltjesGamma[1,(1/3)] *) );


CBHS-2C22//Simplify

CBHS-2C22==Log[2]-EulerGamma//FullSimplify


Clear[m,n];

(1/4) (-4 EulerGamma+Log[(1024/27)]+6 Log[(π/(Gamma[(1/3)] Gamma[(2/3)]))])//FullSimplify

Gamma[(1/3)] Gamma[(2/3)]//FullSimplify

(1/4) (-4 EulerGamma+Log[(1024/27)]+6 Log[(π/(Gamma[(1/3)] Gamma[(2/3)]))]);
%==(1/4) (-4 EulerGamma+10Log[2]-3Log[3]+6 Log[(π/(Gamma[(1/3)] Gamma[(2/3)]))])