Processing math: 100%

Friday, September 23, 2016

Entry 16

Ramanujan gave an unusual approximation to π
π4522ln[(5+292)3(529+116)(9+364+5+364)6]
which is good to 31 digits. This in fact belongs to a family. First, define the fundamental units U2=1+2U29=5+292U58=99+1358U174=1451+110174
Note that 174=6×29. These are involved in fundamental solutions to Pell equations.  For example, for x258y2=1, it is (x,y)=(99,13), (see the values for U58). Then with increasing precision π158ln[26(U29)12]π1258ln[29((U2)3U29U58)6]π1358ln[26(U29)12(U174)2(9+364+5+364)24]π1458ln[29((U2)3U292U58)3(v+1+v)12]
where v=21/2(U2)6(U29)3.

Beautifully consistent, aren't they?  The last is by this author and is accurate to 42 digits. The expression inside the log function is the exact value of (η(τ)η(2τ))24 where  η(τ) is the Dedekind eta function, and  τ=582,τ=2582,τ=3582,τ=4582, respectively.

Anyone can find a nice expression for the next step?  

No comments:

Post a Comment