More about Pi(x) and its derivatives...

[eljose79]

studying the integral equation:

log(R(s)/s=Int(0,infinite)Pi(x)/x(x**s-1) and derivating and itegrating i have got to set an integral equation for dPi(x)/dx but now i wuld like to know if dPi(x)/dx could be expanded into a series of eigenfunctions of the kernel so we could solve it....in fact
dPi(n)/dn=(for big n)=1/l(x)-1/Ln(x)Ln(x) ubt i do not know if this will be enough.

[eljose79]

I forgot to say the interval (2,infinite) in fact if it is not valid for dPi(x)/dx would be valid an expansion for d2Pi(x)/dx2 or higher?..thanks...

[matt grime]

pi is the prime counting function yes? well it only has one sided derivatives, and the derivative is everywhere 1