#Anthony Zaleski hw 6
#OK to post 

with(numtheory):

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

##########
#Problem 1
##########

#CheckZ5(N): checks (V) in Zagier's paper
CheckZ5:=proc(N) local x:
evalf(add((th(x)-x)/x^2,x=2..N)):
end:

##########
#Problem 3
##########

#gT(f,t,z,T): inputs an expression f in a variable t, a complex number z, 
#and a real number T, and outputs the integral of f(t)*exp(-z*t) 
#for t from 0 to T.
gT:=proc(f,t,z,T)
int(f*exp(-z*t),t=0..T):
end:




########C6.txt#######

#C6.txt, Feb. 5, 2018, Experimental Mathematics
Help:=proc(): print(`Z(s,N), Ph(s,N) , th(x) , EP(s,N), CheckZ4(N,alpha,eps),CheckZ5(N) `): end:

with(numtheory):

#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:

#Checks IV in Zagier's paper
CheckZ4:=proc(N,alpha,eps) 

evalf([eps*Ph(1+eps,N), eps*Ph(1+eps+I*alpha,N), eps*Ph(1+eps+I*2*alpha,N)]),
evalf(add(binomial(4,2+r)*Ph(1+eps+I*r*alpha,N),r=-2..2)):

end: