tmp/tmp3r0z8is6/{from.md → to.md}
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| 1 |
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#### Riemann zeta function <a id="sf.cmath.riemann.zeta">[[sf.cmath.riemann.zeta]]</a>
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``` cpp
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double riemann_zeta(double x);
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float riemann_zetaf(float x);
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long double riemann_zetal(long double x);
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```
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*Effects:* These functions compute the Riemann zeta function of their
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respective arguments `x`.
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*Returns:* $$%
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\mathsf{\zeta}(x) =
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\left\{
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\begin{array}{cl}
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\displaystyle
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\sum_{k=1}^\infty k^{-x},
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& \mbox{for $x > 1$}
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\\
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\\
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\displaystyle
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\frac{1}
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{1 - 2^{1-x}}
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\sum_{k=1}^\infty (-1)^{k-1} k^{-x},
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& \mbox{for $0 \le x \le 1$}
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\\
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\\
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\displaystyle
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2^x \pi^{x-1} \sin(\frac{\pi x}{2}) \, \Gamma(1-x) \, \zeta(1-x),
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& \mbox{for $x < 0$}
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\end{array}
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\right.
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\;$$ where x is `x`.
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