Relate the square-lattice identity \[ 60\sum_{\lambda\in\mathbb Z[i]\setminus\{0\}}\lambda^{-4} = 64\pi^2\frac{\Gamma(5/4)^4}{\Gamma(3/4)^4} \] to the conformal radius of the square with side length \(2\), namely \[ \frac{4\Gamma(3/4)^2}{\pi^{3/2}} =1.078705\ldots. \]
Solution
Set \[ g_2 =60\sum_{\lambda\in\mathbb Z[i]\setminus\{0\}}\lambda^{-4} \] and let \(r\) denote the conformal radius of the square with side length \(2\).
1. The square-lattice identity can be written as\[ g_2=\frac{\pi^6}{\Gamma(3/4)^8}. \]
Euler’s reflection formula and the gamma recurrence give \[ \Gamma(1/4)\Gamma(3/4)=\pi\sqrt2 \] and \[ \Gamma(5/4) =\frac14\Gamma(1/4) =\frac{\pi\sqrt2}{4\Gamma(3/4)}. \] Substituting this into the square-lattice identity yields \[ \begin{aligned} g_2 &=64\pi^2 \frac{\Gamma(5/4)^4}{\Gamma(3/4)^4}\\ &=64\pi^2 \frac{\pi^4}{64\Gamma(3/4)^8}\\ &=\frac{\pi^6}{\Gamma(3/4)^8}. \end{aligned} \]
2. The conformal radius satisfies\[ r^4=\frac{256\Gamma(3/4)^8}{\pi^6}. \]
From the stated formula \[ r=\frac{4\Gamma(3/4)^2}{\pi^{3/2}}, \] raising both sides to the fourth power gives \[ r^4 =\frac{4^4\Gamma(3/4)^8}{\pi^6} =\frac{256\Gamma(3/4)^8}{\pi^6}. \]
3. The two identities are related by\[ \boxed{ g_2=\left(\frac4r\right)^4, \qquad\text{equivalently}\qquad r=4g_2^{-1/4}. } \]
Multiplying the identities in 1 and 2 gives \[ g_2r^4=256=4^4. \] Since \(g_2>0\) and \(r>0\), taking the positive fourth root gives the stated relation.
4. The relation in 3 is also the direct link between the two Schwarz–Christoffel integrals.
Let \[ I=\int_1^\infty\frac{dx}{\sqrt{x^3-x}}. \] The square-lattice period calculation in E-HM213A17-151 gives \[ g_2=4I^4. \] Hence \[ 4g_2^{-1/4}=\frac{2\sqrt2}{I}. \] Using \[ I =\frac12B\left(\frac14,\frac12\right) =\frac{\sqrt\pi\,\Gamma(1/4)}{2\Gamma(3/4)} \] and Euler’s reflection formula gives \[ \frac{2\sqrt2}{I} =\frac{4\Gamma(3/4)^2}{\pi^{3/2}} =r. \] Thus the elliptic period determining the square-lattice Eisenstein invariant is the reciprocal-scaled Schwarz–Christoffel integral determining the conformal radius.
5. Q.E.D.
Steps <1>1–<1>3 give the exact algebraic relation, and step 4 identifies the common period integral underlying the two formulas.