On Tue, Jan 25, 2011 at 3:54 AM, Bill Gosper <billgosper@gmail.com> 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
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))
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.
Yes, it's immediate. Let exp(-pi k) =: q. Rewrite eta as its infinite product; log(prod) -> sum(logs); log -> powerseries; sumswap; closed form inner sum. --rwg (c1) [%E^(%PI*K*N) = Q^-N,%E^-(%PI*K) = Q,%PI*K = -LOG(Q)]; %pi k n 1 - %pi k (d1) [%e = --, %e = q, %pi k = - log(q)] n q (c2) SUBST(%,SUM(1/(N*(%E^(%PI*K*N)-1)),N,1,INF) = -LOG(ETA(%E^-(%PI*K)))-%PI*K/24); inf ==== \ 1 log(q) (d2) > ---------- = ------ - log(eta(q)) / 1 24 ==== n (-- - 1) n = 1 n q (c3) MAKEINFPROD(%); inf inf ==== /===\ \ 1 | | i1 (d3) > ---------- = - log( | | (1 - q )) / 1 | | ==== n (-- - 1) i1 = 1 n = 1 n q (c4) NICEINDICES(SUMIFY(%)); inf inf ==== ==== \ 1 \ i (d4) > ---------- = - > log(1 - q ) / 1 / ==== i (-- - 1) ==== i = 1 i i = 1 q (c5) SUBSTPART(POWERSERIES(PIECE,Q,0),%,2,1,1); inf inf inf ==== ==== ==== i i1 + i \ 1 \ \ q (d5) > ---------- = > i > --------- / 1 / / i i1 + i ==== i (-- - 1) ==== ==== i = 1 i i = 1 i1 = 0 q (c6) SUMSWAP(NICEINDICES(%)); inf inf inf ==== ==== ==== j k + k \ 1 \ \ k q (d6) > ---------- = > > ---------- / 1 / / j k + k ==== i (-- - 1) ==== ==== i = 1 i j = 0 k = 1 q (c7) SUBSTPART(HYPERSIMP(PIECE),%,2,1); inf inf ==== ==== j + 1 \ 1 \ q (d7) > ---------- = > -------------------- / 1 / j + 1 ==== i (-- - 1) ==== (j + 1) (1 - q ) i = 1 i j = 0 q (c8) LINCHANGE(%,J-1); inf inf ==== ==== j \ 1 \ q (d8) > ---------- = > ---------- / 1 / j ==== i (-- - 1) ==== j (1 - q ) i = 1 i j = 1 q
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 at http://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