$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

Question

In a paper about Prime Number Races, I found the following (page 14 and 19):

This formula, while widely believed to be correct, has not yet been proved. $$ \frac{\int\limits_2^x{\frac{dt}{\ln t}} - \# \{\text{primes}\le x\} } {\sqrt x/\ln x} \approx 1 + 2\sum_{\gamma} \ \ \frac{\sin(\gamma\ln x)}{\gamma}, \tag{3} $$ with $\gamma$ being imaginary part of the roots of the $\zeta$ function.

$\dots$

For example, if the Generalized Riemann Hypothesis is true for the function $L(s)$ just defined, then we get the formula $$ \frac{\#\{\text{primes}\ 4n+3 \le x\} - \#\{\text{primes}\ 4n+1 \le x\}} {\sqrt x/\ln x} \approx 1 + 2\sum_{\gamma^\prime} \frac{\sin(\gamma^\prime\ln x)}{\gamma^\prime}, \tag{4'} $$ with $\gamma^\prime$ being imaginary part of the roots of the Dirichlet $L$-function associated to the race between primes of the form $4n+3$ and primes of the form $4n+1$, which is $$ L(s) = \frac1{1^s} - \frac1{3^s} + \frac1{5^s} - \frac1{7^s} + \dots. $$

1. Since $$ \begin{eqnarray} \# \{\text{primes}\le x\} &=&\# \{\text{primes}\ 4n+3 \le x\} + \#\{\text{primes}\ 4n+1\le x\}\\ &\approx& \text{Li}(x)- \left(\sqrt x/\ln x\right) \left(1 + 2\sum_{\gamma} \ \ \frac{\sin(\gamma\ln x)}{\gamma} \right) \end{eqnarray} $$ and assuming the (Generalized) Riemann Hypothesis to be true, is it valid to calculate $$ \begin{eqnarray} \# \{\text{primes}\ 4n+3 \le x\} &\approx& \frac{\text{Li}(x)}{2} &-& \frac{\left(\sqrt x/\ln x\right)}{2} \left(1 + 2\sum_{\gamma} \ \ \frac{\sin(\gamma\ln x)}{\gamma} \right)\\ &&&+& \frac{\left(\sqrt x/\ln x\right)}{2} \left( 1 + 2\sum_{\gamma^\prime} \frac{\sin(\gamma^\prime\ln x)}{\gamma^\prime} \right)\\ &\approx& \frac{\text{Li}(x)}{2} &+& \left(\sqrt x/\ln x\right) \left(\sum_{\gamma^\prime} \frac{\sin(\gamma^\prime\ln x)}{\gamma^\prime} -\sum_{\gamma} \ \ \frac{\sin(\gamma\ln x)}{\gamma} \right), \end{eqnarray} $$ or do the error terms spoil the calculation?

2. Is there another way to get $\# \{\text{primes}\ 4n+3 \le x\}$ using a different Dirichlet $L$-function? How does it look like? Is it possible to treat the general case of $\# \{\text{primes}\ kn+m \le x\}$ the same way?

EDIT From the wiki page on Generalized Riemann hypothesis (GRH), I get:

Dirichlet's theorem states that if a and d are coprime natural numbers, then the arithmetic progression a, a+d, a+2d, a+3d, … contains infinitely many prime numbers. Let π(x,a,d) denote the number of prime numbers in this progression which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every coprime a and d and for every ε > 0 $$ \pi(x,a,d) = \frac{1}{\varphi(d)} \int_2^x \frac{1}{\ln t}\,dt + O(x^{1/2+\epsilon})\quad\mbox{ as } \ x\to\infty $$ where φ(d) is Euler's totient function and O is the Big O notation. This is a considerable strengthening of the prime number theorem.

So my example would look like $$ \pi(x,3,4) = \frac{1}{\varphi(4)}\text{Li}(x) + O(x^{1/2+\epsilon}), $$ (something that already ask/answered here: Distribution of Subsets of Primes). So the part with the roots seems to be burried in $O(x^{1/2+\epsilon})$, since $\varphi(4)=2$.

Thanks...

Answer 1

(1) is a correct computation. In general, to treat primes of the form $kn+m$, you would have a linear combination of $\phi(k)$ sums, each of which runs over the zeros of a different Dirichlet $L$-function (of which the Riemann $\zeta$ function is a special case). And yes, assuming the generalized Riemann hypothesis, all of the terms including those sums over zeros can be estimated into the $O(x^{1/2+\epsilon})$ term.

To find out more, you want to look for "the prime number theorem for arithmetic progressions", and in particular the "explicit formula". I know it appears in Montgomery and Vaughan's book, for example.

Answer 2

Here is a table of the 331 first zeros for numerical evaluation (warning: values could be missing...)

6.0209489046975966549025115255,
10.243770304166554552137757473,
12.988098012312422507453109785,
16.342607104587222194976861486,
18.291993196123534838526004279,
21.450611343983460497200948384,
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32.592186527117155130815194045,
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