Upper bounds for the sum of primes up to $n$

By partial summation $$ s(n) = n\pi(n)-\sum_{m=2}^{n-1}\pi(m) $$ so by the Prime Number Theorem $$ s(n) = \frac{n^2}{\log n}-\sum_{m=2}^{n-1}\frac{m}{\log m}+O\left(\frac{n^2}{\log^2 n}\right). $$ The sum on the right is $$ \sum_{m=2}^{n-1}\frac{m}{\log m} = \int_2^n \frac{x}{\log x}dx + O\left(\frac{n}{\log n}\right) $$ using the monotonicity properties of the integrand. Now the integral equals, by partial integration, $$ \int_2^n \frac{x}{\log x}dx = \left[\frac{x^2}{2\log x}\right]_2^n + \int_2^n \frac{x}{2\log^2 x}dx = \frac{n^2}{2\log n} + O\left(\frac{n^2}{\log^2 n}\right).$$ Altogether we have $$ s(n) = \frac{n^2}{2\log n} + O\left(\frac{n^2}{\log^2 n}\right).$$ This can be made more precise both numerically and theoretically.

It is not difficult to calculate upper bounds on $s(n)$ from bounds on the prime counting function $\pi(n)$. Just use integration by parts, $$ s(n) = \int_0^n x\,d\pi(x) = n\pi(n) - \int_0^n\pi(x)\,dx. $$ I'm not sure what the currently best known bounds for $\pi(x)$ are but, checking Wikipedia, gives $$ \frac{x}{\log x}\left(1+\frac{1}{\log x}\right) < \pi(x) < \frac{x}{\log x}\left(1+\frac{1}{\log x}+\frac{2.51}{(\log x)^2}\right) $$ with the left hand inequality holding for $x\ge599$ and the right hand holding for $x\ge355991$. So,

$$ s(n)\le \frac{n^2}{\log n}\left(1+\frac{1}{\log n}+\frac{2.51}{(\log n)^2}\right)-\int^n\left(1+\frac{1}{\log x}\right)\frac{x\,dx}{\log x}+c $$ (where $c$ is a constant which you can compute if you feel so inclined). Applying integration by parts,

$$ s(n)\le\frac{n^2}{2\log n}\left(1+\frac{1}{\log n}+\frac{5.02}{(\log n)^2}\right)-\frac12\int^n\left(1+\frac{2}{\log x}\right)\frac{x\,dx}{(\log x)^2}+c $$

Bounding $\log x\le\log n$ in the integral gives a bound

$$ s(n)\le\frac{n^2}{2\log n}\left(1+\frac{1}{2\log n}+\frac{4.02}{(\log n)^2}\right)+c $$

You can also take $c=0$ if you only require the bound to hold for $n\ge N$ (some $N$), since the term I neglected in the integral by applying $\log x\le \log n$ grows withuot bound, and will eventually dominate any constant term. Obviously, if you know any better bounds for $\pi(n)$ then you will get improved bounds for $s(n)$. For example, the same Wikipedia article linked to above states that $\left\vert\pi(x)-{\rm Li}(x)\right\vert\le\frac{\sqrt{x}\log x}{8\pi}$ for $x\ge2657$ under the assumption that the Riemann hypothesis holds.

There definitely are earlier references than our book. An asymptotic formula for

$\sum_{p \leq x} p^a$

is in T. Salát and S. Znám, On the sums of the prime powers, Acta Fac. Rer. Nat. Univ. Com. Math. 21 (1968), pp. 21-24.