Prime zeta function

In mathematics, the Prime zeta function is an analogue of the Riemann zeta function. It is defined as the analytic continuation to $\mathbb{C}$ of the following infinite series, which converges for $\Re(s) > 1$ :

$P(s)=\sum_{p\,\in\mathrm{\,primes}} \frac{1}{p^s}$

[edit] Integral of the prime zeta function

$\int\sum_{p\,\in\mathrm{\,primes}}\frac{1}{p^s}\;\mathbf{d}s=-\sum_{p\,\in\mathrm{\,primes}}\frac{1}{p^s\log p}+\mathbf{C}$

$\int_{1}^{\infty}\sum_{p\,\in\mathrm{\,primes}}\frac{1}{p^s}\;\mathbf{d}s=\sum_{p\,\in\mathrm{\,primes}}\frac{1}{p\log p}$

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