I am reading the article *D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991)* ([click](https://math.dartmouth.edu/~carlp/PDF/paper82.pdf)).
In equation (27) the authors, apparently, used the following upper bound for a sum over the prime numbers:
$$(\star)\quad\sum\_{p \leq x} p^{-1+\varepsilon} \ll \frac{x^{\varepsilon}}{\varepsilon\log x} ,$$
for sufficiently large $x > 0$ and sufficiently small $\varepsilon > 0$, where the implicit constant in $\ll$ is absolute. (Precisely, $\varepsilon = \frac{4+\log\log\log x}{\log\log x}$, but I hope that this is not really important.) 
I am trying to prove ($\star$) but since now I have failed. Clearly, one way can be using partial summation and the prime number theorem, for example
$$\sum\_{p \leq x} p^{-1+\varepsilon} = \pi(x) x^{-1+\varepsilon} + (1-\varepsilon)\int\_2^x \pi(t)t^{-2+\varepsilon}dt ,$$
but then I am now been able to prove the claim. I also throught that
$$\sum\_{p \leq x} p^{-1+\varepsilon} \leq \sum\_{n \leq \pi(x)} n^{-1+\varepsilon} \ll \int\_0^{\pi(x)} t^{-1+\varepsilon}dt = \frac{\pi(x)^\varepsilon}{\varepsilon} \ll \frac{x^\varepsilon}{\varepsilon (\log x)^\varepsilon} ,$$
but his is too weak.


Thank you in advance for any suggestion.



Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$
The left hand side of $(\star)$ is at least
$$ \sum\_{p\leq x}p^{-1}=\log\log x +O(1), $$
hence a necessary condition for the truth of $(\star)$ is that
$$ \frac{x^{\varepsilon}}{\varepsilon\log x} \gg \log\log x.$$
This condition is also sufficient, in the light of the following bound that I prove below:
$$ (\star\star)\quad\sum\_{p \leq x} (p^{-1+\varepsilon} - p^{-1})\ll \frac{x^{\varepsilon}}{\varepsilon\log x}. $$
In particular, $(\star)$ is valid for $\varepsilon>(\log\log\log x + \log\log\log\log x)/ \log x$, since in this case
$$ \frac{x^{\varepsilon}}{\varepsilon\log x} > \frac{(\log\log x)(\log\log\log x)}{\log\log\log x + \log\log\log\log x} \gg \log\log x.$$


Now let me prove $(\star\star)$:
$$ \sum\_{p \leq x} (p^{-1+\varepsilon} - p^{-1}) \ll \sum\_{p\leq e^{1/\varepsilon}}\frac{\varepsilon\log p}{p}+\int\_{e^{1/\varepsilon}}^x t^{-1+\varepsilon}\,d\pi(t) $$
$$ \ll 1+\frac{x^\varepsilon}{\log x}+\int\_{e^{1/\varepsilon}}^x \frac{t^{-1+\varepsilon}}{\log t}\,dt
= 1+\frac{x^\varepsilon}{\log x}+\int\_1^{\varepsilon\log x}\frac{e^t}{t}dt\,\ll\frac{e^{\varepsilon\log x}}{\varepsilon\log x}=\frac{x^\varepsilon}{\varepsilon\log x}. $$


