One particularity of the Generalized Riemann Hypothesis seems to deserve some clarification. In particular, what is included in the commonly accepted version of the conjecture?


GRH states that 


$$\displaystyle \pi(x; a, d) = \frac{1}{\phi(d)} \int\_2^x \frac{dt}{\log t} + O\left(x^{1/2 + \epsilon}\right),$$


where $\pi(x; a, d)$ is the number of primes $p \leq x$ such that $p \equiv a \pmod{d}$. Here it is assumed that $\gcd(a,d) = 1$. 


My question concerns the implicit constant inherent in the big $O$-notation. In particular, what is conjectured about the nature of this constant? Is it required to depend on $\epsilon$ only or is it required to depend on $d$ (maybe $a$ also?) as well? If it cannot be expected to be uniform in $d$, then what can be conjectured about its dependence on $d$? 


It seems that an explicit version of the GRH where the implicit constant's dependence on $d$ and $\epsilon$ is needed to obtain the necessary strength to obtain even a partial improvement to the Bombieri-Vinogradov theorem, or the Elliot-Halberstam conjecture.



The implicit constant in GRH
You seem to confuse GRH with its consequences (or equivalent formulations). GRH is a statement about the zeros of Dirichlet $L$-functions, and there is no implicit constant in the statement.


At any rate, your displayed formula with any implicit constant depending on $a$ and $d$ implies GRH for Dirichlet $L$-functions. In the other direction, GRH for Dirichlet $L$-functions implies your displayed formula with a good implicit constant, as mentioned by Lucia.


