On Mon, Sep 15, 2014 at 5:28 PM, Bill Gosper <billgosper@gmail.com> wrote:
(Finch, p95), which ISC teases is -1/2 - i ( 5*Zeta[3]/2 - 6*EulerGamma/5 - 9*Log[3]/5 -1.52145483*10^-8) We need a few more digits to rule this out. We can probably get them from
Victor Miller>Also, the following paper "Zeta Expansions of Classical Constants" (for some reason never published): http://algo.inria.fr/flajolet/Publications/landau.ps <
This is a bit hard to find, even under its current title "Zeta Function Expansions of Classical Constants". (Firefox absquatulated before I could grab URL.)
But caution: Eqn (10) is almost certainly a copy&paste blunder from eqn (6). I.e., -1/1^m - ... - 1/(n0-1)^m should be -1/p_1^m - ... - 1/p_{n0-1}^m .
Indeed, the correct formula is In[364]:= Sum[f[1/Prime[n]], {n, \[Infinity]}] -> Sum[f[1/Prime[n]], {n, m}] + Sum[Derivative[k][f][0]/k!*(PrimeZetaP[k] - Sum[1/Prime[n]^k, {n, m}]), {k, 2, \[Infinity]}] In[366]:= %364 /. m -> 1 should speed it up to ≥ 1 bit/term. Out[366]= 1 Sum[f[--------], {n, Infinity}] -> Prime[n] -k (k) 1 (-2 + PrimeZetaP[k]) f [0] f[-] + Sum[------------------------------, {k, 2, Infinity}] 2 k! Note that to get, e.g., f[1/Prime[n]] := 1/Prime[n]/(Prime[n]+1), you need f->Composition[1/#/(#+1)&,1/#&] In[563]:= Evaluate[Simplify[1/(# (# + 1)) &@(1/# &)@#]] & Out[563]= #1^2/(1 + #1) & In[563]:=%366 /. f -> %563 Out[564]= 1 Sum[------------------------, {n, Infinity}] -> 1 2 (1 + --------) Prime[n] Prime[n] 1 - + Sum[ 6 2 -k #1 (k) (-2 + PrimeZetaP[k]) (------ & ) [0] 1 + #1 ----------------------------------------, k! {k, 2, Infinity}] In[565]:= N[{%[[1]] /. \[Infinity] -> 9999, %[[2]] /. \[Infinity] -> 33}, 22] Out[565]= {0.3302291628777476319724, 0.3302299262642031960435} In[566]:= N[%%[[2]] /. ∞-> 34, 22] Out[566]= 0.3302299262642032560056 In[567]:= N[%%%[[2]] /. ∞ -> 35, 22] Out[567]= 0.3302299262642032360182 A few more terms or a slightly larger m will provide several dozen good digits. The formula isn't general enough to compute the twin primes constant Product[1 - 1/(Prime[n] - 1)^2, {n, 2, ∞}] but the general technique in the paper, namely expand the summand in powers of Prime[n] and then sumswap, produces "prime zeta" functions which are well supported in Mathematica. So I got lots of digits agreeing with the published value, but for a factor of 2 which I'm not bothering to track down. So with PrimeZetaP we can crank out a bunch of precise values from slowly convergent sums and products over primes and look with the usual tools for unsuspected numerical relations. --rwg
On Sun, Jun 2, 2013 at 10:58 AM, Mike Stay <metaweta@gmail.com> wrote: http://mathworld.wolfram.com/PrimeProducts.html Lots of fun stuff here, like prod(i=1, inf, p_i^s) = (2pi)^2s
On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote: [Alan Adler, of Aeropress fame]
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
Consciously or not, I'll bet Steampunk was inspired by Hanna-Barbera's Flintstones. --rwg