On Tue, Jan 25, 2011 at 7:44 AM, Oleksandr Pavlyk <pavlyk@wolfram.com>wrote:
On 1/25/2011 5:54 AM, Bill Gosper wrote:
Holy bleep! It seems that SUM(1/(N*(%E^(%PI*K*N)-1)),N,1,INF) = -LOG(ETA(%E^-(%PI*K)))-%PI*K/24 inf ==== \ 1 - %pi k %pi k
----------------- = - log(eta(%e )) - ----- / %pi k n 24 ==== n (%e - 1) n = 1
Bill, assuming that I understood your notations correctly, the above is not true:
In[64]:= With[{k = 7`50}, {bSum[k], -Log[DedekindEta[I k/2]] - Pi k/24}]
Out[64]= {7.92010695079811226453151563444882317242882923932*10^-20, 2.814268458673568764008084010947351043967*10^-10}
Curiously, these are almost square roots of each other:
In[65]:= Sqrt[First[%]]/Last[%]
Out[65]= 0.9999999995778597315091688742994744685907
Hi Sasha, this is strange. Macsyma gets
(c10) (fancy_display:false,SUM(1/(N*(%E^(%PI*K*N)-1)),N,1,INF) = -LOG(ETA(%E^-(%PI*K)))-%PI*K/24) inf ==== \ 1 - %pi k %pi k (d10) > ----------------- = - log(eta(%e )) - ----- / %pi k n 24 ==== n (%e - 1) n = 1 (c11) Block([fpprec:33],apply_nouns(bfloat(subst([k=7,inf=print(ceiling(log(10)*fpprec/7/%pi))],%)))) 4 (d11) 2.81426845867356876400808401094735b-10 = 2.81426845867356876400808921254564b-10 with the inaccuracy due to serious cancellation on the rhs: (c12) bfloat(subst(k=7,args(rhs(d10)))) (d12) [9.1629785757844987375b-1, - 9.1629785729702302789b-1] So it is not our usual problem of disparate eta notations. E.g.,
SUM(1/(N*(%E^(%PI*N/SQRT(3))-1)),N,1,INF) =
-3*LOG(GAMMA(1/3))/2+LOG(%PI)+LOG(SQRT(3)+1)/4-3*LOG(3)/8+9*LOG(2)/8-%PI/(24*SQRT(3))
This statement also does not appear to be true Bill.
In[84]:= N[{bSum[1/Sqrt[3]], -(3/2) LogGamma[1/3] + Log[\[Pi]] + Log[Sqrt[3] + 1]/4 - (3 Log[3])/8 + (9 Log[2])/8 - \[Pi]/( 24 Sqrt[3])}, 50]
Out[84]= {0.027665600934284027762918972514225264544812434292230, \ 0.21009804651298308433103368320796309488706292986618}
Again, (c14) Block([fpprec:33],apply_nouns(bfloat(subst([k=1/sqrt(3),inf=print(ceiling(log(10)*fpprec/(1/sqrt(3))/%pi))],%)))) 42
(d14) 2.10098046512983084331033683207963b-1 = 2.10098046512983084331033683207963b-1 But PSLQ easily finds an error:
In[87]:= coeffs = FindIntegerNullVector[{N[bSum[1/Sqrt[3]], 500], LogGamma[1/3], Log[Pi], Log[1 + Sqrt[3]], Log[2], Log[3], Pi/Sqrt[3]}]
Out[87]= {-24, -36, 24, 0, 32, -9, -2}
In[88]:= ExpandAll[ Solve[{bSum[1/Sqrt[3]], LogGamma[1/3], Log[Pi], Log[1 + Sqrt[3]], Log[2], Log[3], Pi/Sqrt[3]} . coeffs == 0, bSum[1/Sqrt[3]]]]
Out[88]= {{bSum[ 1/Sqrt[3]] -> -(Pi/(12*Sqrt[3])) + (4*Log[2])/3 - (3*Log[3])/8 + Log[Pi] - (3/2)*LogGamma[1/3]}}
--Sasha
My only guess is that bSum is not what we think. --Bill
inf 1
==== 3 log(gamma(-)) \ 1 3 log(sqrt(3) + 1) 3 log(3) 9 log(2) %pi
----------------- = - --------------- + log(%pi) + ---------------- - -------- + -------- - ---------- / %pi n 2 4 8 8 24 sqrt(3) ==== ------- n = 1 sqrt(3) n (%e - 1)
(so I misspoke about the 3/4 log(pi)-log(gamma(1/4)), which only appears for rational k). This can't be hard.
S. Plouffe>Hello mr. Gosper,
now that is interesting, you have the explicit formula for sum(1/n/(exp(Pi*n*k)-1),n=1..infinity) when k=1,2,3,4,5,6.
rwg>In principle, k = sqrt(any rational).
SP> Here with PSLQ or LLL I could get 1,2 and 4 only separatly in terms of log(Pi), Pi, log(2) and log(GAMMA(1/4)),
I did not know that one could get those explicit algebrico-log-gamma expressions . What is surprising is the<cancel out> of the algebraic expression when k = 1/5, 2/5 and 4/5 and also the approximations when k = 2/13, 2/7 or 2/163,
I tried to find other fractions for k and found only that explicit 1/5, 2/5 and 4/5. Nevertheles, I have found a simpler formulation of the formula for pi ; it appears on my home page athttp://www.plouffe.fr/
<http://www.plouffe.fr/>rwg>Ah, but you still show only the approximate value of
Sum[n^3/(E^(2*Pi*n/7)-1),{n,Infinity}==
1/4 3/4 1 8 (301 + 210 Sqrt[2] 7 + 120 Sqrt[7] + 90 Sqrt[2] 7 ) Gamma[-] 1 4 -(---) + ------------------------------------------------------------------ 240 6 5120 Pi
(that quadrinomial probably factors some.)
sp> I should switch to macsyma! best regards, Simon Plouffe
I'm afraid the best hope for Macsyma at this point is for Noftsker to get it out of probate and the Maxima guys swallow their pride and cannibalize as much as possible. --rwg