I just got SUM(N/(%E^(2*SQRT(11)*%PI*N)-1),N,1,INF) = -SQRT(22^(1/3)*(2733*SQRT(33)+624899)^(1/3)+22^(1/3)*(624899-2733*SQRT(33))^(1/3)+240)*GAMMA(1/22)*GAMMA(3/22)*GAMMA(5/22)*GAMMA(9/22)*GAMMA(15/22)/(8448*SQRT(11)*%PI^(7/2))-1/(8*SQRT(11)*%PI)+1/24
Joerg Arndt has just sent me a 2004 paper (THANKS!) by J. Yi actually deriving closed forms for theta[3](0,q) for q = +-exp(-n pi) for n as large as 12, in terms of <algebraic>*pi^(1/4)/Gamma(3/4). This is my first inkling that this eta stuff I've been bombarding you with isn't new. (Eavesdropper Bruce Berndt submitted the paper. Bruce, can you tell me what's known about Theta/Eta special values? (Not quotients.))
I got eta(exp(-pi Sqrt(43))) as a nice algebraic times a *horrible* pile of Gamma(k/86)^(n/10), for nearly all 0<k<43. The logs of all those Gammas resist numerical relation-finding. Those tenth roots are news, too.
*Were* news. I got it about half as long with the empirical identity 2^(21/43) Pi^(3/2) Gamma[1/86] Gamma[3/86]^3 Gamma[5/86] Gamma[7/86]^2 Gamma[ 9/86] Gamma[11/86]^5 Gamma[13/86] Gamma[15/86]^2 Gamma[ 17/86] Gamma[19/86]^3 Gamma[21/86] Gamma[23/86]^2 Gamma[ 25/86] Gamma[27/86]^4 Gamma[29/86] Gamma[31/86]^2 Gamma[ 33/86] Gamma[35/86]^3 Gamma[37/86] Gamma[39/86]^2 Gamma[41/86] == Sqrt[43] Gamma[1/43]^2 Gamma[2/43] Gamma[3/43]^4 Gamma[ 4/43] Gamma[5/43]^2 Gamma[6/43] Gamma[7/43]^3 Gamma[ 8/43] Gamma[9/43]^2 Gamma[10/43] Gamma[11/43]^6 Gamma[ 12/43] Gamma[13/43]^2 Gamma[14/43] Gamma[15/43]^3 Gamma[ 16/43] Gamma[17/43]^2 Gamma[18/43] Gamma[19/43]^4 Gamma[ 20/43] Gamma[21/43]^2 This is presumably a nonobvious consequence of the ntuplication formula, presumably involving intermediate swell, since FullSimplify can't find it. There's probably a 43 <- n generalization that would be worth finding. E.g., 2^(20/41) Sqrt[41] Gamma[1/41]^4 Gamma[2/41] Gamma[3/41]^2 Gamma[4/41] Gamma[5/ 41]^3 Gamma[6/41] Gamma[7/41]^2 Gamma[8/41] Gamma[9/41]^6 Gamma[10/ 41] Gamma[11/41]^2 Gamma[12/41] Gamma[13/41]^3 Gamma[14/41] Gamma[ 15/41]^2 Gamma[16/41] Gamma[17/41]^4 Gamma[18/41] Gamma[19/ 41]^2 Gamma[20/41] == Pi Gamma[1/82]^3 Gamma[3/82] Gamma[5/ 82]^2 Gamma[7/82] Gamma[9/82]^5 Gamma[11/82] Gamma[13/82]^2 Gamma[ 15/82] Gamma[17/82]^3 Gamma[19/82] Gamma[21/82]^2 Gamma[23/ 82] Gamma[25/82]^4 Gamma[27/82] Gamma[29/82]^2 Gamma[31/82] Gamma[ 33/82]^3 Gamma[35/82] Gamma[37/82]^2 Gamma[39/82] But there are lots of these if we repeatedly drop the smallest arg and adjoin a new largest: (Sqrt[41] Gamma[1/41] Gamma[2/41] Gamma[3/41]^2 Gamma[4/41] Gamma[5/ 41]^3 Gamma[6/41] Gamma[7/41]^2 Gamma[8/41] Gamma[9/41]^6 Gamma[ 10/41] Gamma[11/41]^2 Gamma[12/41] Gamma[13/41]^3 Gamma[14/ 41] Gamma[15/41]^2 Gamma[16/41] Gamma[17/41]^4 Gamma[18/41] Gamma[ 19/41]^2 Gamma[20/41] Gamma[21/41]^3)== (4 2^(18/41) Pi^(5/2) Gamma[3/82] Gamma[5/82]^2 Gamma[7/82] Gamma[9/82]^5 Gamma[11/ 82] Gamma[13/82]^2 Gamma[15/82] Gamma[17/82]^3 Gamma[19/82] Gamma[ 21/82]^2 Gamma[23/82] Gamma[25/82]^4 Gamma[27/82] Gamma[29/ 82]^2 Gamma[31/82] Gamma[33/82]^3 Gamma[35/82] Gamma[37/ 82]^2 Gamma[39/82]) So it seems like any set of half the fractions in (0,n)/n will form a "basis", and by judiciously introducing fractions >1/2, it is likely possible to significantly reduce the number of different Gamma species. One of those n/2 choose n ought to be nice. Testing the shorter eta identity: In[117]:= {N[%], N[%, 33]} Out[117]= {False, True} Right helpful those Booleanizations. They drive my solving to Macsyma. --rwg
The Lambert series can't be much better: different algebraic times same Gammas to different powers.
Not so bad, now.
--rwg
Eye mercy: http://gosper.org/newetas.html
Updated.
Does anybody really want to see eta(exp(-pi Sqrt(43)))? If you have Mma, DedekindEta[I*Sqrt[43]] == ((-(80/(1 + 63*Sqrt[129])^(1/3)) + (1 + 63*Sqrt[129])^(1/3))^(1/8)* Gamma[7/43]^(1/5)*Gamma[11/43]^(9/10)*Gamma[15/43]^(7/10)* (Gamma[4/43]*Gamma[6/43]*Gamma[10/43]*Gamma[14/43]* Gamma[16/43])^ (2/5)*(Gamma[3/43]*Gamma[19/43])^(3/5)* ((Gamma[1/86]*Gamma[9/86]*Gamma[13/86]*Gamma[17/86]* Gamma[21/86]* Gamma[25/86]*Gamma[41/86])/(Gamma[2/43]*Gamma[8/43]* Gamma[12/43]* Gamma[18/43]*Gamma[20/43]))^(1/10)* ((Gamma[1/43]*Gamma[5/43]*Gamma[9/43]*Gamma[13/43]* Gamma[17/43]* Gamma[21/43])/(Gamma[7/86]*Gamma[15/86]*Gamma[23/86]* Gamma[31/86]*Gamma[39/86]))^(3/10))/(2*86^(7/40)* Pi^(17/20)* Sqrt[Gamma[11/86]]*(Gamma[3/86]*Gamma[19/86])^(7/10)* Gamma[27/86]^(3/5)*Gamma[35/86]^(1/5)* (Gamma[5/86]*Gamma[29/86]*Gamma[33/86]*Gamma[37/86])^(2/5))
Working toward sqrt(163)pi.
It's threatening to be hideous. --rwg
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float!
But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho.
I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg