# On the least prime in arithmetic progressions

My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that $$p(a, q) \ll q^L$$ for some absolute constant $L$. Wiki page for this theorem lists a number of papers that estimate $L$ with the most recent result by Xylouris who proved that $L \leq 5.2$.

It is also known that the Generalized Riemann Hypothesis implies $$p(a, q) \ll (q\log q)^2 \text{,}$$ while in 1978, Heath-Brown conjectured even tighter bound: $$p(a, q) \ll q(\log q)^2 \text{.}$$ I'm wondering whether this last bound, if true (it is still an open problem), implies something non-trivial about $L$-functions?

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I'd guess that you could try to plug in the relevant values into the explicit formula (for example see equation (2) here: math.ubc.ca/~gerg/teaching/613-Winter2011/LinnikTheorem.pdf) and do the computations, but I'll leave a more authoritative statement on this to the experts. –  Timothy Foo Mar 8 '12 at 7:02

I think I agree. If you knew there were lots of small primes in every arithmetic progression - essentially the desired asymptotic number with a small error term - then that would probably improve the known zero-free region for Dirichlet $L$-functions, up to a proof of the generalized Riemann hypothesis if the error term were good enough. But just one small prime in each residue class, I'm not sure that would give us any leverage. –  Greg Martin Mar 13 '12 at 18:41