On Wed, Jun 5, 2013 at 2:25 AM, Bill Gosper <billgosper@gmail.com> wrote:
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
E.g, In[10]:= FunctionExpand[Zeta'[2]] Out[10]= 1/6 \[Pi]^2 (EulerGamma + Log[2] - 12 Log[Glaisher] + Log[\[Pi]]) [...]
Numerically testing, In[7]:= N[Product[Prime[n]^(Prime[n]^2 - 1)^-1, {n, \[Infinity]}]] - E^-(Zeta'[2]/Zeta[2]) During evaluation of In[7]:= Prime::intpp: Positive integer argument expected in Prime[15.]. >> During evaluation of In[7]:= Prime::intpp: Positive integer argument expected in Prime[14.]. >> During evaluation of In[7]:= Prime::intpp: Positive integer argument expected in Prime[13.]. >> During evaluation of In[7]:= General::stop: Further output of Prime::intpp will be suppressed during this calculation. >> Out[7]= -0.0084166 Children should be shielded from software that claims 15. is not a positive integer. A stronger test: NLimit[Product[Prime[n]^(Prime[n]^2 - 1)^-1, {n, oo}], oo -> \[Infinity]] - E^-(Zeta'[2]/Zeta[2]) returned ~10^-7 after a couple of days. I had selected the output, but not copied it into the clipboard, when I switched notebooks for a trivial search. The third time I clicked Next, the whole front end crashed. --rwg