(Repeated from below:) Caution: Product[Prime[n]^(1/(Prime[n]* (3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine. WRI just informed me they fixed this in Mma 10. I don't trust this transcription, but Mod[Prime[n], 3] 1/(1 + (-1) Prime[n]) Product[Prime[n] , EulerGamma 4 16 E Pi {n, oo}] -> ------------------ 1/4 1 6 3 3 Gamma[-] 3 seems to work mathematically (with crash-provoking ∞ for oo). Is this known to converge? I doubt Glaisher's derivation provides proof. This suggests a bunch of convergence puzzles of the form Sum (-1)^Axxx(n)/f(n), where Axxx is some integer sequence and f is some slowly growing function like n or log(1+n) or log(1+log(1+n)) ... . Presumably a sufficiently gradual f could betray bunchiness mod 2 in something like Mod[Prime[n],3] = A039701 <http://oeis.org/A039701>. Glaisher used a master theorem: In[280]:= Product[n^(Sin[2*n*\[Pi]*\[Mu]]/n),{n,\[Infinity]}]->(Sqrt[Sin[\[Pi]*\[Mu]]]*\[CapitalGamma][\[Mu]]/(2^(1/2-\[Mu])*\[Pi]^(1-\[Mu])*E^((1/2-\[Mu])*EulerGamma)))^\[Pi] Notice Mma's evaluation of the lhs: Out[280]= E^(-(1/2) I ((PolyLog^(1,0))[1,E^(-2 I \[Pi] \[Mu])]-(PolyLog^(1,0))[1,E^(2 I \[Pi] \[Mu])]))->(2^(-(1/2)+\[Mu]) E^(-EulerGamma (1/2-\[Mu])) \[Pi]^(-1+\[Mu]) Sqrt[Sin[\[Pi] \[Mu]]] \[CapitalGamma][\[Mu]])^\[Pi] That's the derivative wrt the *order* of the polylog, of whose numerics it seems capable. --rwg On Mon, Jun 3, 2013 at 12:41 PM, Bill Gosper <billgosper@gmail.com> wrote:
Ah, later, during the derivation, Glaisher clarifies: 1 2 1/6 1/12 1/18 1/30 1/36 6 EllipticK[- (2 - Sqrt[3])] 7 13 19 31 37 4 -------------------------------- ... = ----------------------------- 1/6 1/12 1/18 1/24 1/30 EulerGamma 2 5 11 17 23 29 E Pi
where EllipticK[1/4 (2 - Sqrt[3])] == (3^(1/4) Gamma[1/3]^3)/(4 2^(1/3) π)
(Apparently, K was considered a special value of Gamma whenever possible.)
Note how Mma automatically collects all the twin primes: Rule @@ %152 /. oo -> 14
43 1/42 31 1/30 19 1/18 13 1/12 7 1/6 1/36 (--) (--) (--) (--) (-) 37 41 29 17 11 5 ------------------------------------------------- -> 1/24 23
1 2 6 EllipticK[- (2 - Sqrt[3])] 4 ----------------------------- EulerGamma 2 E Pi
Convergence is less smooth than this would suggest--even nonmonotonic, I think. Which explains why yesterday's NLimit of the infinite product is still running. The bunchiness of the primes is nonlethal:
In[126]:= NLimit[Product[1 - Prime[k]^-2, {k, n}], n -> ∞]
Out[126]= 0.607927
In[127]:= % - 6/π^2
Out[127]= -2.39218*10^-9
took a couple of minutes. --rwg
On Sun, Jun 2, 2013 at 8:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
Unfortunately, Glaisher's very next example is a product from "p=2" to "p=∞" of an f(s,p), "where p is any prime, and s = +1 or -1 according as p ≡ 1 or ≡ 2 mod 3". What's s(3)?? Numerically, it seems
Product[Prime[n]^(1/(-1 + (3 - 2*Mod[Prime[n], 3])*Prime[n])), {n, 1, ∞}] == (3*3^(5/8)*Gamma[1/3]^6)/(E^EulerGamma*(16*Pi^4)),
although NLimit lingers in bovoparturition.
Note that for these products to converge, the number of 3k+1 primes <n must stay roughly even with the # of 3k+2 as n->∞. Likewise 4k+1 and 4k+3. --rwg
On Sun, Jun 2, 2013 at 1:55 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow, I just found a smearoxed 241(!) page paper,
http://digreg.mathguide.de/cgi-bin/ssgfi/anzeige.pl?db=reg&ci=QJPAM&id=ART&s...
On products and series involving
prime numbers only, by the indefatigable JWL Glaisher, Sc.D., FRS, in Vol XXVII of the Quarterly Journal of Pure and Applied Mathematics, (which Glaisher co-edited), 1895. The *most preliminary* result, on page 1: Product[Prime[n]^(1/(-1 + I^(-1 + Prime[n])*Prime[n])), {n, 2, ∞}] == Gamma[1/4]^4/(E^EulerGamma*4*Pi^3) --rwg Caution: Product[Prime[n]^(1/(Prime[n]*(3 - 2*Mod[Prime[n], 3]) - 1)), {n, 2,∞}] -> 2^(2/3)*3*EllipticK[Sin[π/12]^2]^2/π^2/E^EulerGamma instantly blows the Mma 9.0.x math engine.
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg