Large prime divisors in small intervals

Question

For my thesis I would like to find integers (lying in a certain residue class) in small intervals which have large prime divisors. And for some reason I decided that I want all bounds appearing in my thesis to be explicit, so I am looking for something like the following result:

Given a residue class $a \pmod{m}$ and an integer $x \ge c_0$ (where $c_0 = c_0(m)$ may depend on $m$, but is bounded above by some explicit function of $m$), there exists an integer $n \equiv a \pmod{m}$ in the interval $[x, x + x^{c_1}]$ having a prime divisor larger than $n^{c_2}$.

Now, I don't really care about the constants $c_1$ and $c_2$, as long as $c_2 > c_1$.

The theorem in this paper by Ramachandra is more or less what I need, except for the restriction on the residue class and the fact that it's not explicit. On the other hand, theorem 1 in this paper by Laishram and Shorey gives the above with $n^{c_2}$ replaced by $\frac{2}{m}x^{c_1}$ for $m \ge 3$, $x \ge 19$.

Does anyone have a reference (or proof) for me?

Full disclosure: I asked this question here two weeks ago, without much success, so I decided to try my luck here.

Answer 1

You can simplify Ramachandra's method by bounding the last sum of p.305 using Brun-Titchmarsh inequality (or Montgomery & Vaughan error-term free version of it) instead of Van der Corput's method + Selberg's sieve. This yields $c_0(m) = e^{k m}$ for some explicit constant $k$ (I am not going to explicit $k$, though !), and for the exponents $$ c_1 = \frac{1}{2} + \frac{\varphi(m)}{4m}, \quad \quad c_2 = \frac{1}{2} + \frac{\varphi(m)}{4m} + \frac{\varphi(m)^2}{80m^2}. $$ If you restrict to $m$ prime (or almost prime), these exponents are essentially independent of $m$.