[Helge von Koch](http://link.springer.com/article/10.1007%2FBF02403071) proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound 
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$



> 
> Q1: Is von Koch's result (and proof) also valid for the generalized Riemann hypothesis (GRH)?
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That is, if a and d are coprime, and $\pi(x,a,d)$ denotes the number of primes in the arithmetic progression $a, a+d, a+2d, \dots$, is GRH equivalent to 
$$
\left|\pi(x,a,d)-\frac{1}{\varphi(d)}\textrm{li}(x)\right|=O(\sqrt{x} \log x)?
$$
The [Wikipedia page for GRH](http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis) states that GRH implies this error term, but is short on references and says nothing about whether the given error term implies GRH. 



> 
> Q2: Does anyone know if [von Koch's paper](http://link.springer.com/article/10.1007%2FBF02403071) has been translated to English; alternatively a reference to the best modern version of proving his result; or perhaps both? 
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Is there a von Koch-type theorem for the generalized Riemann hypothesis?
As GH from MO and Felipe Voloch have already indicated it is standard to show that $\psi(x;q,a) = x/\phi(q) +O(x^{\frac 12+\epsilon})$ for all reduced residue classes $a\pmod q$ is equivalent to GRH for the characters $\pmod q$. I want to make the following small (but amusing) refinement: it is enough to know that $\psi(x;q,1) = x/\phi(q) + O(x^{\frac12+\epsilon})$ and $\psi(x) =x+ O(x^{\frac 12+\epsilon})$ and from these two pieces of information (rather than all $\phi(q)$ residue classes) we can get GRH for all the characters $\pmod q$. 


To see this, note that (starting with Re$(s)>1$) 
$$ 
 -\frac{1}{\phi(q)} 
\sum\_{\chi\neq \chi\_0} \frac{L^{\prime}}{L}(s,\chi) = \int\_1^{\infty} \frac{s}{x^{s+1}} \Big( \psi(x;q,1) - \frac{\psi(x,\chi\_0)}{\phi(q)} \Big) dx, 
$$ 
and so by hypothesis, the RHS extends analytically to Re$(s)>1/2$. Therefore $L(s,\chi) \neq 0$ for all $\chi \neq \chi\_0$ and Re$(s)>1/2$. Finally $\psi(x)=x+O(x^{\frac 12+\epsilon})$ implies RH; thus GRH follows for all the characters $\pmod q$.


Put differently, this shows that the asymptotic $\psi(x;q,1) = \psi(x)/\phi(q) + O(x^{\frac 12+\epsilon})$ implies $\psi(x;q,a) =\psi(x)/\phi(q)+ O(x^{\frac 12+\epsilon})$ for all $(a,q)=1$. 


