hash values [@Heule2013]. Even larger hash values are needless in practice, because it is unrealistic to encounter cardinalities of order $2^{64}$.

Derivation of the Raw Estimator {#sec:derivation_raw_estimator}
-------------------------------

To better understand why the raw estimator fails for small and large cardinalities, we start with a brief and simple derivation without the restriction to large cardinalities ($\symCardinality\rightarrow\infty$) and without using complex analysis as in [@Flajolet2007]. Assume that the register values have following cumulative distribution function $$\label{equ:assumed_register_val_distribution}
\symProbability(\symRegValVariate \leq \symRegVal\vert\symPoissonRate) = e^{-\frac{\symPoissonRate}{\symNumReg 2^{ \symRegVal}}}.$$ For now we ignore that this distribution has infinite support and differs from the register value distribution under the Poisson model , whose support is limited to $[0, \symRegRange+1]$. For a random variable $\symRegValVariate$ obeying the expectation of $2^{-\symRegValVariate}$ is given by $$\label{equ:expectation_power_two}
\symExpectation(2^{-\symRegValVariate})
=
\frac{1}{2}
\sum_{\symRegVal = -\infty}^\infty
2^{-\symRegVal}
e^{-\frac{\symPoissonRate}{\symNumReg 2^{\symRegVal}} }
=
\frac{\symAlpha_\infty\,\symNumReg\,\symPowerSeriesFunc\!\left(\log_2\!\left(\symPoissonRate/\symNumReg\right)\right)}{\symPoissonRate},$$ where the function $$\symPowerSeriesFunc(\symX):= \log(2) \sum_{\symRegVal = -\infty}^\infty
2^{\symRegVal+\symX}
e^{-2^{\symRegVal+\symX}}$$ is a smooth periodic oscillating function with mean 1 and an amplitude that can be bounded by $\symEpsPowerSeriesFunc:=\num{9.885e-6}$ as shown in \[fig:power\_series\_func\]. This limit can also be found using Fourier analysis [@Ertl2017].

![The deviation of $\symPowerSeriesFunc(\symX)$ from 1.[]{data-label="fig:power_series_func"}](power-series-function-minus-1){width="\columnwidth"}

For a large ($\symNumReg\rightarrow \infty$) sample $\symRegValVariate_1,\ldots,\symRegValVariate_\symNumReg$, that is distributed according to , we asymptotically 