hi

so I've been trying for awhile to write an asymptotic formula for

sum p_n<=x 1/(p_n - p(n-1))

(the sum of reciprocals of prime gaps)

where p_n means nth prime and n>1.

my attempt is this

x/ln(x)^2*(A*ln(ln(X))+(B*C_twin))+err.term, C_twin=0.66016...

where A=approx 0.37787 and B is Brun's constant.

The value of A is ln(2*C_twin)+1/10, and the error term is very small even for large x. Can someone verify this please? I can send you a spreadsheet if you like, I've tested this up to 500,000..

thanks from Guy