#Okay to post
#Yonah Biers-Ariel, Feb 11th, Assignment 6

with(numtheory):

#1. CheckZ5(N) approximates the integral in (V) in Zagier's paper. 
CheckZ5:=proc(N) local i:
add(evalf((th(i)-i)/i^2),i=1..N):
end:

#3. gT(f,t,z,T) finds the integral of f(t)*exp(-z*t) for t from 0 to T
gT:=proc(f,t, z,T):
evalf(Int(f*exp(-z*t),t=0..T)):
end:


#Z(s,N)
Z:=proc(s,N) local n:
add(1/n^s,n=1..N):
end:

#Ph(s,N): sum(log(p)/p^s, first N primes)
Ph:=proc(s,N) local i:
add(  log(ithprime(i))/ithprime(i)^s,i=1..N):
end:

#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:

#EP(s,N): the product of 1/(1-1/p^s)) over the first N primes
EP:=proc(s,N) local i:
expand(mul(1/(1-1/ithprime(i)^(s)),i=1..N)):
end: