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