[math-fun] Re^2: Plouffe's 2 surprising approximations and 1 exact formula for pi.
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. 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
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
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} 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
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
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
Yes, Bill, you were right. bSum computed Sum[1/((-1 + E^(2 n*Pi*k))*n), {n, Infinity}] In[1]:= Clear[bSum]; bSum[k_Real] := Module[{sum = 0, prev, n = 1, pwr, term = 1}, pwr = Exp[Pi k]; While[True, prev = sum; term *= pwr; sum += 1/((term - 1)*n); If[sum - prev == 0, Break[]]; n++ ]; sum] In[3]:= With[{k = 7`1000}, bSum[2 k] - bSum[2/k] == (Log[k]/2 - Pi*(k - 1/k)/12)] // AbsoluteTiming Out[3]= {0.1080000, True} In[4]:= With[{k = 1/3`1000}, DedekindEta[I k/2] == Exp[-Pi k/24] QPochhammer[Exp[-Pi*k]]] Out[4]= True Then the other formula also holds as you claimed: In[6]:= 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[6]= {24, 36, -24, -6, -27, 9, 1} In[7]:= 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[7]= {{bSum[1/Sqrt[3]] -> -(\[Pi]/(24 Sqrt[3])) + (9 Log[2])/8 - ( 3 Log[3])/8 + 1/4 Log[1 + Sqrt[3]] + Log[\[Pi]] - 3/2 LogGamma[1/3]}} Given quick convergence of bSum, this gives a quick algorithm for computing Gamma[ k / 3 ] for integer values of k. For what other rational argument r can LogGamma[ r ] be computed using the like of bSum ? --Sasha On 1/25/2011 2:45 PM, Bill Gosper wrote:
On Tue, Jan 25, 2011 at 7:44 AM, Oleksandr Pavlyk <pavlyk@wolfram.com <mailto: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/>
<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
Hello, I think all rational values are feasible in terms of the Gamma function : let me explain, since the rhs of sum(1/n/(exp(Pi*n*a/b)-1),n=1..infinity) = a product of gammas, then by taking the log we should get a linear combination of log(GAMMA(p/q)), p and q being well chosen, The main difficulty is that for small a/b like 2/7 then the degree of the algebraic portion of the gamma's could be of a very high degree, that does mean : p and q have to be chosen in a wide range of values, so even with PSLQ, LLL (the one you like most) it won't give the general solution but only small and easy cases. already with the case of 2/7 computed by Bill Gosper, is already of a fairly high degree, at least 28 ? Also, a technical detail : the sum is something like 1+ 240 *something, of course the <1> has to be removed before taking the log, if not : it won't work, it has to be a pure product of GAMMA(p/q) values. In other words for values like 2/13, or foolishly 2/163 that one is just plain impossible to get with that method in my opinion. also, (to bill gosper), I will change my home page to include your formula for 2/7 and if there is enough place the case 2/13 as well. I am preparing a paper about that , it should come out soon, and with proper credit of course, best regards, simon plouffe
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
participants (3)
-
Bill Gosper -
Oleksandr Pavlyk -
Simon Plouffe