I am looking for the lower and upper bounds for the sum of the first $n$ primes, $S(n)$ for all $n \geq 2$
I found that $S(n) = \frac{n^2}{2 \log n} + O(\frac{n^2}{\log^2n})$. This is only an upper bound, and also not explicit.
What are some explicit (i.e without asymptotic notation) lower and upper bounds for $S(n)$ that fit all $n \geq 2$?
EDIT
I just found good bounds from this paper https://arxiv.org/abs/1606.06874 (Axler 2016):
A lower bound for all $n \geq 2$ And an upper bound for all $n \geq 1897700$.
There is also an upper bound for all $n \geq 6$ (from a different source):
$S(n) \leq \frac{p_n^2}{\log{p_n}} \leq \frac{n^2(\log n + \log \log n)^2}{\log n(\log n + \log \log n -1)}$
and we can refactor it for $n \geq 2$
$S(n) \leq \frac{p_n^2}{\log{p_n}} \leq \frac{n^2(\log n + \log \log n)^2}{\log (n(\log n + \log \log n -1))}+6$
Is there a better upper bound for $n \geq 2$?
