It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any $\varepsilon>0$, $\pi(n)\gg n^{1-\varepsilon}$.


Chebyshev famously proved that if $\lim \pi(n)\log n$ exists it must be equal to 1, but I seem to remember that he also proved bounds on that value, pushing the date back to 1850 or so in that case. But were there earlier results in this direction?



Who first proved that there are at least n^(1-ε) primes up to n?
In the 1850's, Chebyshev gave the following explicit bound, for sufficiently large n.


$$0.92129 \, \frac{n}{\log n} < \pi(n) < 1.0556 \, \frac{n}{\log n}.$$ 


This is mentioned in "An introduction to the theory of the Riemann zeta function" by S. J. Patterson. There is an exercise in the first chapter that gives the lower bound following Chebyshev's method (that is, using Stirling's formula and the series $\log \ n! = \Sigma\_{p^k\leq n} \lfloor n/p^k \rfloor\log p$). S. J. Patterson cites Chebyshev as the first to obtain significant results toward the prime number theorem.


