- Besides the approach in Section 1.2, there are several alternate ways of dealing with the sum \(\textstyle \sum { 1 } / ( z + \omega ) ^ { 2 }\) , where \(\omega = n + m \tau\) . For example, one may sum either (a) circularly, (b) first in n then in m, (c) or first in m then in n.
- Prove that \({ \mathrm { i f ~ } } z \notin \Lambda\) , then
\[ \lim _ {R \to \infty} \sum_ {n ^ {2} + m ^ {2} \leq R ^ {2}} \frac {1}{(z + n + m \tau) ^ {2}} = S _ {1} (z) \]
exists and \(S _ { 1 } ( z ) = \wp ( z ) + c _ { 1 }\)
- Similarly,
\[ \sum_ {m} \left(\sum_ {n} \frac {1}{(z + n + m \tau) ^ {2}}\right) = S _ {2} (z) \]
exists and \(S _ { 2 } ( z ) = \wp ( z ) + c _ { 2 } , { \mathrm { w h e r e ~ } } c _ { 2 } = F ( \tau )\) , and F is the forbidden Eisenstein series.
- Also
\[ \sum_ {n} \left(\sum_ {m} \frac {1}{(z + n + m \tau) ^ {2}}\right) = S _ {3} (z) \]
exists with \(S _ { 3 } ( z ) = \wp ( z ) + c _ { 3 }\) , and \(c _ { 3 } = \tilde { F } ( \tau )\) , the reverse of \(F ,\)
[Hint: To prove (a), it sufices to show that lim \(. R { \longrightarrow } \infty\) \(\sum { \begin{array} { r l } \end{array} } \quad 1 / ( n + m \tau ) ^ { 2 } = c _ { 1 }\) \(1 \leq n ^ { 2 } + m ^ { 2 } \leq R ^ { 2 }\) exists. This is proved by a comparision with \(\begin{array} { r } { \int _ { 1 \leq x ^ { 2 } + y ^ { 2 } \leq R ^ { 2 } } \frac { d x } { ( x + y \tau ) ^ { 2 } } = I ( R ) } \end{array}\) . It can be shown that \(I ( R ) = 0\) , which follows because \(( \overline { { x } } + y \tau ) ^ { - 2 } = - ( \partial / \partial x ) ( x + y \tau ) ^ { - 1 } . ]\)
Solution
Write \(\Lambda=\{n+m\tau:n,m\in\mathbb Z\}\), with \(\Im\tau\ne0\). Recall \[ \wp(z)=\frac1{z^2}+\sum_{\omega\in\Lambda\setminus\{0\}} \left(\frac1{(z+\omega)^2}-\frac1{\omega^2}\right). \tag{1} \] The series in parentheses converges absolutely and locally uniformly away from \(\Lambda\), because its summand is \(O_z(|\omega|^{-3})\). Hence any admissible summation convention for the raw series differs from \(\wp(z)\) only by the same convention applied to \(\sum_{\omega\ne0}\omega^{-2}\).
For circular summation, put \[ C_R=\sum_{1\le n^2+m^2\le R^2}\frac1{(n+m\tau)^2}. \] We show \(C_R\) converges. Since \(\Im\tau\ne0\), there is \(c>0\) such that \[ |x+y\tau|\ge c\sqrt{x^2+y^2}. \] Set \(\phi(x,y)=(x+y\tau)^{-2}\). On the unit square centered at \((n,m)\), with \(n^2+m^2\) large, the mean-value theorem gives \[ |\phi(x,y)-\phi(n,m)|\le C(n^2+m^2)^{-3/2}. \] The sum of these errors over \((n,m)\ne(0,0)\) converges, since the number of lattice points with radius comparable to \(r\) is \(O(r)\).
Thus \(C_R\) differs by a convergent quantity from the integral of \(\phi\) over the corresponding union of unit squares. Replacing that union by the annulus \[ 1\le x^2+y^2\le R^2 \] changes the integral by \(O(R^{-1})\), because only an \(O(R)\)-area boundary layer is changed and \(|\phi|=O(R^{-2})\) there. But \[ I(R)=\int_{1\le x^2+y^2\le R^2}\frac{dx\,dy}{(x+y\tau)^2}=0. \] Indeed, in polar coordinates, \[ I(R)=\log R\int_0^{2\pi}\frac{d\theta}{(\cos\theta+\tau\sin\theta)^2}, \] and the angular integral is zero because \[ \frac{d}{d\theta} \left(\frac{\sin\theta}{\cos\theta+\tau\sin\theta}\right) =\frac1{(\cos\theta+\tau\sin\theta)^2}. \] (The denominator never vanishes because \(\Im\tau\ne0\).) Therefore \(C_R\to c_1\), and by (1) \[ S_1(z)=\wp(z)+c_1. \]
For summation first in \(n\), define \[ F(\tau):=\sum_{m\in\mathbb Z}\left(\sum_{n\in\mathbb Z}^{\!*}\frac1{(n+m\tau)^2}\right), \] where the star omits \((n,m)=(0,0)\). For \(m=0\), the inner sum is \(2\zeta(2)=\pi^2/3\). For \(m\ne0\), \[ \sum_{n\in\mathbb Z}\frac1{(n+m\tau)^2} =\pi^2\csc^2(\pi m\tau). \] Because \(|\csc(\pi m\tau)|=O(e^{-\pi|m|\,|\Im\tau|})\), the outer sum converges absolutely. Thus \(F(\tau)\) exists, and absolute convergence of the corrected series in (1) gives \[ S_2(z)=\wp(z)+F(\tau). \] This is the forbidden Eisenstein series.
Similarly, summing first in \(m\), \[ \widetilde F(\tau) :=\sum_{n\in\mathbb Z}\left(\sum_{m\in\mathbb Z}^{\!*}\frac1{(n+m\tau)^2}\right). \] For \(n\ne0\), factor out \(\tau^{-2}\): \[ \sum_{m\in\mathbb Z}\frac1{(n+m\tau)^2} =\tau^{-2}\pi^2\csc^2\!\left(\frac{\pi n}{\tau}\right), \] while the \(n=0\) term is \(\tau^{-2}\pi^2/3\). Since \(\Im(-1/\tau)>0\) when \(\Im\tau>0\), these terms again decay exponentially in \(|n|\). Hence \(\widetilde F(\tau)\) converges and \[ S_3(z)=\wp(z)+\widetilde F(\tau). \] Thus all three summation procedures exist, and they differ only by the constants produced by the corresponding summations of \(\sum_{\omega\ne0}\omega^{-2}\).