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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.

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    $\begingroup$ 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. $\endgroup$ – Lucia Dec 30 '13 at 3:20
  • $\begingroup$ 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? $\endgroup$ – Stanley Yao Xiao Dec 30 '13 at 13:58
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    $\begingroup$ 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. $\endgroup$ – Joël Dec 30 '13 at 15:50
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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.

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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.

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  • $\begingroup$ Where can one find a proof of this result? $\endgroup$ – Joël Jan 2 '14 at 3:35
  • $\begingroup$ Bach and Shallit give this reference: J. Oesterlé, Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisé, Astérisque 61 (1979). Unfortunately, I think this is only an announcement, and that the promised proofs never appeared in print. $\endgroup$ – so-called friend Don Jan 2 '14 at 4:31

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