#C5.txt, Feb. 1, 2018, ExpMath (Spring 2018, RU)

Help:=proc(): print(` vM(n), CheckI18a(n), CheckI18(N), Sx(x) , CheckI24(x), psix(x),CheckI25half(x)  `): end:

with(numtheory):

#vM(n): The von-Mangoldt function log(p) is n=p^a 0 otherwise
vM:=proc(n) local a:

if n=1 then
  RETURN(0):
fi:

a:=ifactors(n)[2]:
if nops(a)>1 then
 0:
else
 log(a[1][1]):
fi:
end:

#CheckI18a(n): Checks (1.18) that log(n)=Sum(vM(d), d in divisors of n)
CheckI18a:=proc(n) local S,d:
S:=divisors(n):

simplify(log(n)-add(vM(d), d in S)):
end:

#Sx(x): sum of log(n) n<=x
Sx:=proc(x) local n:
add(log(n),n=1..trunc(x)):

end:

#CheckI24(x):Checks Eq. (2.4)
CheckI24:=proc(x)
evalf((Sx(x)-x*log(x)+x)/log(x)):
end:






CheckI18:=proc(N)  local n:
evalb({seq(CheckI18a(n), n=1..N)}={0}):
end:

#psix(x): the famous psi(x): sum of log(p) over all prime powers p^a<=x
 psix:=proc(x) local n:
 add(vM(n),n=1..x):
end:

 CheckI25half:=proc(x) local m:
  simplify(Sx(x)-add(psix(x/m),m=1..x)):
 end: