$L^p$ Asymptotics of the M\"obius Energy Density of Helix Curves

Motivated by the recent work of Lipton on the M"obius energy of helix curves, we extend the study to the $L^p$ asymptotics of the meromorphic family $$M_{\rho }(t)=\frac{{\rho }^2+1}{{\rho }^2{t}^2+4{\sin }^2(t\mathrm{/}2)}-\frac{1}{t^2}\mathrm{.}$$ The helix has infinite M"obius energy, but the arclength-rescaled energy density is finite. As $\rho \to 0$ the helix coils infinitely tight. Using contour integration and a careful Laurent expansion near the poles, we establish $I_p(\rho ):={({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{M}_{\rho }(t{)}^{p}dt)}^{1\mathrm{/}p}\sim C_p{\rho }^{-(2-1\mathrm{/}p)}$ for integer $p>1$, extended to real $p>1$, where $C_p$ is an explicit constant involving $\zeta (2p-1)$. The result gives the precise $L^p$ blowup rate of the M"obius energy density as the pitch $\rho \to 0$. The borderline case $p=1$ yields a logarithmic correction $I_1(\rho )\sim \log (1\mathrm{/}\rho )\mathrm{/}\rho$, recovering Lipton’s main theorem. We derive a quantitative coiling barrier. Numerical verification confirms the scaling exponent to high precision.