This will be the $5$th in a series of an infinite series of a single function. Here are 2 related sums:

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> [A Kelvin-Bessel Sum: $$\mathrm{\sum\limits\_{\Bbb N} ker(x)+i\ kei(x)= \sum\limits\_1^\infty K\_0\left(\sqrt ix\right)= 0.133691752… - 0.7256312077… i}$$](https://math.stackexchange.com/questions/4253802/closed-form-of-mathrm-sum-limits-bbb-n-kerxi-keix-sum-limits-1-in)
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and

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> [On $$\mathrm{\sum\limits\_{n=0}^\infty \left(C(n)-\frac{\sqrt\pi}{2\sqrt2}\right)+ \sum\limits\_{n=0}^\infty \left(S(n)-\frac{\sqrt\pi}{2\sqrt2}\right)}$$](https://math.stackexchange.com/questions/4223393/on-mathrm-sum-limits-n-0-infty-leftcn-frac-sqrt-pi2-sqrt2-right)
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Our goal sum uses the [Complementary Error function](https://functions.wolfram.com/GammaBetaErf/Erfc/) integrating term by term. Note the [Fresnel Integrals](https://en.m.wikipedia.org/wiki/Fresnel_integral). There are also [**Gamma type functions**](https://functions.wolfram.com/GammaBetaErf/):

$$\sum\_{\Bbb N^0}\text{erfc}(x)=\sum\_{x=0}^\infty \text{erfc}(x) =\lim\_{n\to\infty} \left(n-\sum\_0^n \text{erf}(x)\right)= \frac{2}{\sqrt \pi}\sum\_{x\in \Bbb N^0}\int\_x^\infty e^{-t^2} dt= \sum\_{x\in \Bbb N^0}\left(1-\frac 2\pi \int\_0^\infty \frac{\sin(2tx)}{te^{t^2}}dt\right)=\sum\_{x\in\Bbb N^0}\left(1 - (1 + i) \left(\text C\left(\frac{((1 - i) z)}{\sqrt\pi}\right) - i\,\text S\left(\frac{((1 - i) z)}{\sqrt\pi}\right)\right)\right)=\sum\_{\Bbb N^0}\text{erf}(x,\infty)= \frac{1}{\sqrt \pi}\sum\_{\Bbb N^0}Γ\left(\frac12,x^2\right)= \sum\_{\Bbb N^0}Q\left(\frac12,x^2\right) =\frac{1}{\sqrt \pi} \sum\_{\Bbb N^0}x \text E\_\frac12 \left(x^2\right)= \frac{1}{\sqrt \pi}\int\_1^\infty t^{-\frac12}\sum\_{x\in\Bbb N^0}xe^{-tx^2}dt =1.16199904794712636353230832245579717…$$

As @JohnBarber found:

$$\sum\_{\Bbb N^0}\text{erfc}(x) =1+\frac{2}{\sqrt\pi}\int\_1^\infty \lfloor x\rfloor e^{-x^2} dx$$
The final integral-sum representation reminds me of a [**Differentiated Jacobi Theta function of the Third Kind**](https://functions.wolfram.com/EllipticFunctions/EllipticThetaPrime3/02/). Here is a [question](https://math.stackexchange.com/questions/3252623/evaluate-the-series-sum-k-1-infty-k-e-k2?noredirect=1) about it although there are others.

How can I evaluate the constant? Please correct me and give me feedback!