hihi, all - i transformed rkg's sum first, since (if i'm using the right phi function = euler totient) phi(n) = sum(d|n) mu(n/d) d sum(n>=1) phi(n)/2^n = sum(n>=1, d>=1, d|n) mu(n/d) d / 2^n (here we take m / n/d) = sum(m>=1, d>=1) mu(m) d / 2^(m d) (here we use sum(d>=1) d x^d = x/(1-x)^2, with x = 2^(-m)) = sum(m>=1) mu(m) 2^(-m)/(1 - 2^(-m))^2 which converges very quickly (after m>=53, the values don't change in simple double precision arithmetic) M sum to M 51 1.367630801985022293 53 1.367630801985022070 55 1.367630801985022070 which agrees well enough with rkg's number (to 15 places or so) 1.367630801985022350790508146 well, to check the rest of the digits, i then wrote out some completely ad hoc multiple-precision functions and got (at varying precisions) 1. 3676 3079 9663 1. 3676 3080 1985 0223 0400 1. 3676 3080 1985 0223 5079 0509 2651 1. 3676 3080 1985 0223 5079 0508 1462 1311 4322 1. 3676 3080 1985 0223 5079 0508 1462 1308 8139 0748 9199 8962 7951 8188 1. 3676 3080 1985 0223 5079 0508 1462 1308 8139 0748 9199 8962 7948 5295 6598 4637 6215 6711 9392 which agrees with rkg's numbers as far as they go more later, cal