= Half-integer zeta-regularized mode sum
{title2=$\sum_{n\geq0}^{\mathrm{reg}}(n+\tfrac12)=\tfrac1{24}$}
For $r=n+1/2$, $n\geq0$, the convergent frequency zeta function, expressed through the <Riemann zeta function>, is $\sum_r r^{-s}=(2^s-1)\zeta_R(s)$ when $\operatorname{Re}s>1$. Its <analytic continuation> at $s=-1$ gives $(1/2-1)(-1/12)=1/24$. A common exponential frequency cutoff gives $\sum_r r e^{-\varepsilon r}=\varepsilon^{-2}+1/24+O(\varepsilon^2)$, so the same finite part follows by subtracting the leading divergence. In contrast, integer frequencies have regulated sum $-1/12$.
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