The implicit constant in GRH

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 usual argument (see e.g. Davenport's book) gives that on GRH, $|\pi(x;q,a) - \text{li}(x)/\phi(q)| \le C x^{\frac 12}\log(qx)$ for an absolute constant $C$. GRH doesn't give anything substantially stronger than Bombieri-Vinogradov. – Lucia Dec 30 '13 at 3:20
• If this is the result, why is the error term in the form of GRH stated in the body of my question $O(x^{1/2 + \epsilon})$? Would it not be sharper to have $O(x^{1/2}\log x)$ instead? – Stanley Yao Xiao Dec 30 '13 at 13:58
• Yes, it would evidently be sharper. Actually, it is classically known that all these variants for the error term are equivalent : $O_{d}(x^{1/2} log(x))$, $O(x^{1/2} \log(dx))$, in $O_{d,\epsilon}(x^{1/2+\epsilon})$, etc. They indeed all implies GRH, which as GH recalls is a statement about the zeros of Dirichlet L-function, and conversely, this statement implies the form of the error term given above. Besides Davenport's very nice book, I believe that reading chapter 5 of Iwaniec-Kowalski can be helpful. – Joël Dec 30 '13 at 15:50

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.
Under GRH, one has the completely explicit estimate $$\left|\pi(x;q,a) - \frac{1}{\varphi(q)} \int_{2}^{x} \frac{dt}{\log{t}}\right| \leq \sqrt{x} \log(q^2 x)$$ for all $x\ge 2$ and all choices of coprime $a$ and $q$ with $q\ge 2$. Bach and Shallit attribute this estimate to Oesterle, as a special case of his results on effective Chebotarev.