Here's a nice one (a page or two later): Product[Prime[n]^(1/(Prime[n]^2 - 1)), {n, ∞]}]==E^-(Zeta'[2]/Zeta[2]) . Unfortunately, Mma turns this into a big mess involving the useless symbol Glaisher, which is about as handy as if it arbitrarily gronked zeta[3] to be Apery. --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, 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