#Please do not post homework 
#Ahsan Khan.

#Problem 1:

Th := proc(n) local x:
add(log(x), x in primelist2(n)):
end:

CheckZ5 := proc(N) local x:
evalf(add((Th(n)-n)/n^2, n=1..N)):
end:

#Indeed it seems to converge to. For instance, it reads -2.58 for N=2000.

#Problem 3:

Lap := proc(f,t,z,T) local x:
int(subs(t=x,f)*exp(-z*x), x=0..T);
end: 

#Gives the right Laplace transform for various test functions.