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With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots of Riemann's $\zeta$ resp. Dirichlet $\beta$ function: \begin{align*} \pi^*(x;4,3)&=\sum_{k=0}^\infty 2^{-k-1}\left( \operatorname{R}(x^{1/2^{k}})-\sum_{\rho_\zeta} \operatorname{R}(x^{\rho_\zeta/2^k}) +\sum_{\rho_\beta} \operatorname{R}(x^{\rho_\beta/2^k}) \right) \end{align*}

Ilya helped a lot to derive a formula for summing over General Functions of Primes $$ \sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1} $$ and applying this to Prime $\zeta$ function this simplifies (haha) to $$ P_{x;4,1}(r)+P_{x;4,3}(r)= \sum_{p<\color{red}x} \frac{1}{p^{ir}} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} \left[ {\rm li}(t^{\frac zn-ir}) \right]^{\color{red}x}_2 $$

So I think it's possible to combine these two partial results and come up with something

$$ P_{x;4,3}(r)=\sum_{k=0}^\infty 2^{-k-1} \sum_{n=1}^{\infty}\frac{ \mu (n)}{n}\sum_{z\in\{1,\rho(\zeta),\rho(\beta) \}/2^k}\alpha(z) \left[ {\rm li}(t^{\frac zn-ir}) \right]^{\color{red}x}_2,\tag{2} $$ where $\alpha(z)=\cases{\phantom{-} 1 ,\text{if $z=1/2^k$ or $z=\rho_\beta/2^k$},\\ -1 , \text{if $z=\rho_\zeta/2^k$}}$.

But I also found another, pretty simple, way to represent $P_{x;4,3}(r)$: $$ P_{x;4,3}(r)= \sum_{p<x} \frac12\left(1+ie^{i2\pi\frac{p}4}\right)p^{ir}, $$ where $\left(1+ie^{i2\pi\frac{p}4}\right)$ just cancels for primes of the form $4n+1$. We have to apply $(1)$ and insert $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho(\zeta)}\operatorname{R}(x^{\rho(\zeta)}) \tag{3} $$ to get something comparable to $(2)$, but the roots of Dirichlet's $\beta$ function wouldn't obviously show up here.

Is it possible to show that the effect of $\rho(\beta)$ in $(2 )$ can be condensed to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$?

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    $\begingroup$ the explicit formula for $L(s,\chi_4) = \sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$ is exactly the same as for $\zeta(s)$ : $\frac{L'(s,\chi_4)}{L(s,\chi_4)} = \sum_{\beta} \frac{1}{s-\beta}$ where $\beta$ are the zeros (trivial and non-trivial) of $L(s,\chi_4)$. Hence $\sum_{p^k < x} (-1)^{(p-1)/2} \ln p =\ln(L(0,\chi)) -\sum_{\beta} \frac{x^\beta}{\beta}$ and $2\sum_{p < x, p \equiv 3 \bmod 4} 1 = li(x)- \sum_{\rho} \frac{li(x^\rho)}{\rho} + \sum_{\beta} \frac{li(x^\beta)}{\beta}+\ln L(0,\chi)+\ln \zeta(0)$ $\endgroup$ – reuns Sep 17 '16 at 15:56

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