Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the logarithm integral $x$ , I computed $\int\_{0}^{1} (\pi(x)-Li (x)) $ I [have got](https://www.wolframalpha.com/input/?i=integral%28%28Pi%28x%29-li%28x%29%29%29%2C+x%3D0+to+1) $\log 2$, this result attracted my attention to the behavior of $\int\_{0}^{t} (\pi(x)-Li (x))dx $ , such that i can't juge weither it is divergent or convergent integral , But my question here is:



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> **Question**: What can this $\int\_{0}^{t} ( \pi(x)-Li (x)) $ tell us about primes distribution ?
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What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?
The integral does not converge. See <https://academic.oup.com/blms/article-abstract/31/4/424/277640?redirectedFrom=fulltext>.


On the other hand, a proof that this integral is less than $K.t^{\alpha}$ almost everywhere (i.e. on a set of the form $(t\_{0},\infty)\setminus J$ where $J$ has Lebesgue measure $0$) for some $\alpha<2$ and an absolute constant $K$ would imply that there exists some $1/2\leq\sigma\_{0}<1$ such that $\zeta$ does not vanish in the right half plane $\Re(s)>\sigma\_{0}$, which quantifies the "randomness" of the distribution of the primes: the less $\sigma\_{0}$, the less random this distribution.


