Can we prove that infinite many primes begin with any given digitstring? With Dirichlet's theorem, we can easily prove that infinite many primes end with a given digitstring with final digit $1,3,7$ or $9$. 

Can we also prove that infinite many primes begin with a given digit-string with first digit non-zero ?

Intuively, this should be the case, but I wonder whether we can show it rigorously. I thought of prime gaps, but I am not sure whether the best proven prime gaps are sufficient.

By the Prime Number Theorem, for any fixed $\epsilon>0$, we have  $\pi((1+\epsilon)n)-\pi(n)\to\infty$ as $n\to\infty$. In particular, for any number $a$ representing the desired initial digits, we will find many primes between $a10^k$ and $(a+1)10^k$ for all large enough $k$.
