nt.number theory – An integral involving the function Chebyshev $ psi $

Consider Chebyshev $ psi $ a function $ psi (x) = sum_ {p ^ r leq x} log p $ about the main powers. To define $ I = int_ {1} ^ { infty} ( psi (x) -x) x ^ {- 2} mathrm {d} x $. My question is, do I exist? Someone suggested not to do it, and he puts forward the following: Suppose $ I $ exist. C & # 39; is, $ I = c $ or $ c $ is a constant. then

begin {align}
c = lim_ {M rightarrow infty} Bigg ( int_ {1} ^ {M} ( psi (x) -x) x ^ {- 2} mathrm {d} x Bigg) & = lim_ {M rightarrow infty} Bigg ( int_ {1} ^ {M} psi (x) x ^ {-2} mathrm {d} x- log M Bigg).
end {align}
Let $ M = log N_k $, or $ N_k = p_ {1} p_ {2} cdots p_k $ denotes the product of the first $ k prime numbers. Putting this in the above gives
begin {equation}
c = lim_ {k rightarrow infty} Bigg ( int_ {1} ^ { log N_k} psi (x) x ^ {-2} mathrm {d} x- log log N_k Bigg ).
end {equation}
We know that
$ log log N_k = e ^ {- gamma} prod_ {p leq p_k} (1-p ^ {- 1}) ^ {- 1} + O (1) $ and $ zeta (1 + it) = prod_ {p} (1-p ^ {- 1 it}) ^ {- 1} $ for all true $ t neq $ 0. So leaving $ t rightarrow 0 ^ { pm} $we find that $ prod_ {p leq p_k} (1-p ^ {- 1}) ^ {- 1} rightarrow lim_ {s rightarrow 1 ^ {+}} zeta (s) $ as $ k rightarrow infty $ Therefore $ log log N_ {k} rightarrow e ^ {- gamma} lim_ {s rightarrow 1 ^ +} zeta (s) + O (1) $ as $ k rightarrow infty $. From our first equation, we observe that
begin {equation}
int_ {1} ^ { log N_k} psi (x) x ^ {-2} mathrm {d} x rightarrow – lim_ {s rightarrow 1 ^ +} frac { zeta & # 39; } { zeta} (s)
end {equation}
as $ k rightarrow infty $. By putting these estimates in the previous equation, we obtain
begin {equation}
c = lim_ {s rightarrow 1 ^ +} Bigg (- frac { zeta} { zeta} (s) -e ^ {- gamma} zeta (s) + O (1) ) Bigg).
end {equation}
As $ s rightarrow 1 ^ + $, we know that $ – frac { zeta} { zeta} (s) rightarrow (s-1) ^ {- 1} – gamma $ and $ zeta (s) rightarrow (s-1) ^ {- 1} + gamma $. Inserting these in the yields above
$ c = lim_ {s rightarrow 1 ^ +} Big ( frac {1-e ^ {- gamma}} {s-1} + O (1) Big), $ which is absurd. This implies that our assumption must be false?

$ gamma = 0.57221 cdots $ is the constant of Euler.