# 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.