# Ok to post Homework
# Matt Charnley, Assignment 6, February 11, 2018

with(numtheory):

Help := proc() print(`th(x), IntZ5(n), CheckZ5(N), gT(f, t, z, T)`): end:

#1) Carry out the integration on each interval of length 1, and then add them up.

#th(x): sum(log(p), p<=x)
th:=proc(x) local i,su:
su:=0:

for i from 1 while ithprime(i)<=x do
 su:=su+ log(ithprime(i)):
od:
su:
end:


# IntZ5(n): Computes the integral in Z5 from n to n+1 using the value of th(n)

IntZ5 := proc(n) local x:
evalf(Int((th(n) - x)/x^2, x=n..n+1)):
end:

# CheckZ5(N): Does the integral in V5 from 1 to N to try to determine if it is convergent

CheckZ5:= proc(N) local n:
add(IntZ5(n), n=1.. N-1):
end:

# It looks like this is converging towards something like -2.29 as N -> infinity

#2) I have read it.

#3) 

gT:= proc(f,t,z,T)
evalf(Int(f*exp(-z*t), t=0..T)):
end:

# Based on the definition, if $f$ is integrable, the integral with z = 0 should converge, and it will converge to the integral of f from 0 to infinity
#
# This was confirmed by my experimentation in Maple. All polynomials gave increasing numbers or infinity, e^-t went to 1, and sin(t)/t went to pi/2.