Help:=proc(): print(`ZNt(N,t) , ZNtR(N,t) , FindZ(f,t,T,res), OS(f,t,P), ZI(f,t,P,eps) `): end:

# zeta_N(1/2+I*t)*zeta_N(1/2-I*t)
ZNt:=proc(N,t) local n:
Re(add(1/n^(1/2+I*t),n=1..N)*add(1/n^(1/2-I*t),n=1..N));
end:


#add(1/n,n=1..N) + add(n1^(1/2+I*t)*n2^(1/2-I*t)= ADD(sqrt(n1*n2)* 2(cos(log(n1)-log(n2))*t),1<=n1<n2<=N)
#n^(I*t)=exp(log(n)*I*t)
# zeta_N(1/2+I*t)*zeta_N(1/2-I*t): w/o I
ZNtR:=proc(N,t) local n1,n2:
add(evalf(1/n1),n1=1..N)+2*add(add( evalf(1/sqrt(n1*n2)*cos((log(n2)-log(n1))*t))    , n2=n1+1..N), n1=1..N):
end:


#FindZ(f,t,T,res): inputs a non-negative function f of t, and a pos. nunber T
#and a resolution, and finds all leftmosts of intervals where it changes signs
FindZ:=proc(f,t,T,res) local i,ej,L:
ej:=evalf([seq(subs(t=res*i,f),i=0..trunc(T/res))]):

L:=[]:

for i from 2 to nops(ej)-1 do
  if ej[i]*ej[i+1]<0 then
   L:=[op(L),[(i-1)*res,i*res]]:
  fi:
od:

L:

end:

#OS(f,t,P): inputs a function f of t and an interval P outputs the subinterval where it changes sign
OS:=proc(f,t,P):

if evalf(subs(t=P[1],f)*subs(t=P[2],f))>0 then
FAIL:
elif evalf(subs(t=P[1],f)*subs(t=(P[1]+P[2])/2,f))<0 then
 [P[1],(P[1]+P[2])/2]:
else
  [(P[1]+P[2])/2,P[2]]:
fi:
end:


#ZI(f,t,P,eps)
ZI:=proc(f,t,P,eps) local P1:

P1:=P:
while P1[2]-P1[1]>eps/10 do
 P1:=OS(f,t,P1):
od:
(P1[1]+P1[2])/2:
end: