## Alternative summation orders for the Eisenstein series $$G_2$$ and Weierstrass $$\wp$$-function.(English)Zbl 1455.30004

The authors study the Eisenstein $$G_2$$ series $G_2(\tau)=\sum_{m}\left[\sum_{n}\,\frac{1}{(n+m\tau)^2}\right],$ with $$m,n\in\mathbb{Z},\,(m,n)\not= (0,0)$$, and the Weierstrass $$\wp$$-function $\wp(z)=\frac{1}{z^2}+\sum_{(m,n)\in\mathbb{Z}^2\setminus \{(0,0)\}}\,\left[\frac{1}{(z+n+m\tau)^2}-\frac{1}{(n+m\tau)^2}\right]$ $$\tau\in\mathbb{H},z\not\in \mathbb{Z}\tau+\mathbb{Z}$$, in the context of conditionally convergent series and their behavior under change in the order of summation.
To this end they introduce a set of shapes as follows.
Let $$\mathcal{K}$$ be the class of compact sets $$K\in\mathbb{R}^2$$ that are convex, have non-empty interior, and are symmetric about the $$x$$- and $$y$$-axes. Define:
1. For each $$K\in\mathcal{K}$$, let $$h_k$$ be the real-valued function whose graph is the upper boundary of the shape $$K$$. The functions $$h_k$$ is compactly supported on an interval $$[-A,A]$$, is an even function, and its reflection $$-h_k$$ is the lower boundary of $$K$$.
2. For a shape $$K\in\mathcal{K}$$ and an array $$(a_{m,n})_{m,n\in\mathbb{Z}}$$ of complex numbers, define $\sum_K\,a_{m,n}=\lim_{\lambda\rightarrow\infty}\,\sum_{(m,n)\in(\lambda K)\cap\mathbf{Z}^2}\,a_{m,n},$ provided the limit exists. The authors refer to this sum as the $$K$$-summation, or the shape summation with respect to the shape $$K$$, of the array $$(a_{m,n})$$.
3.If $$K\in\mathcal{K}$$, denote by $$G_2(K,\tau)$$ the $$K$$-summation of the weight-2 Eisenstein series, defined by $G_2(K,\tau)=\sum_{K}\,\frac{1}{(m\tau +n)^2},$ provided the limit defining the summation exists, and with the convention that $$a_{0,0}=0$$.
4. If $$K\in\mathcal{K}$$ and $$G_2(K<\tau)$$ is defined, the authors denote by $$E(K,\tau)$$ the residual function associated to $$K$$, defined as $E(K,\tau)=G_2(K,\tau)-G_2(\tau).$
The main results are the following.
Theorem 5. For all $$\tau\in\mathbb{H}$$ and all $$K\in\mathcal{K}$$, the limit defining $$G_2(K,\tau)$$ exists. The residual function is given by $E(K,\tau)=4\int_0^A\,\frac{h_K(x)}{\tau^2x^2-h_K^2(x)}dx,$ where, as before, $$A$$ denotes a number for which $$h_K$$ is supported on $$[-A,A]$$.
Proposition 9. For $$K\in\mathcal{K},\tau\in\mathbb{H}$$ and $$z\not\in \mathbb{Z}\tau +\mathbb{Z}$$, the $$K$$-summation $$\wp(K,\tau)$$ is defined and satisfies $\wp(K,z)=\wp(z)+G_2(\tau)+E(K,\tau).$

### MSC:

 30B99 Series expansions of functions of one complex variable 40A05 Convergence and divergence of series and sequences
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### References:

  T. M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics 41, Springer, 1990. · Zbl 0697.10023  P. Bruin and S. Dahmen, “Modular forms”, lecture notes, 2016, http://www.few.vu.nl/ sdn249/modularforms16/notes.pdf.  H. Cohn, “A conceptual breakthrough in sphere packing”, Notices Amer. Math. Soc. 64:2 (2017), 102-115. · Zbl 1368.52014  G. Eisenstein, Mathematische Werke, Band I, Chelsea, New York, 1975. · Zbl 0339.01018  I. Grattan-Guinness, The development of the foundations of mathematical analysis from Euler to Riemann, MIT Press, Cambridge, MA, 1970. · Zbl 0215.04401  P. Rosenthal, “The remarkable theorem of Lévy and Steinitz”, Amer. Math. Monthly 94:4 (1987), 342-351. · Zbl 0628.40001  R. Roy, Elliptic and modular functions from Gauss to Dedekind to Hecke, Cambridge University Press, 2017. · Zbl 1401.11002  W. Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill, New York, 1976. · Zbl 0346.26002  E. M. Stein and R. Shakarchi, Complex analysis, Princeton Lectures in Analysis 2, Princeton University Press, 2003. · Zbl 1020.30001  M. S. Viazovska, “The sphere packing problem in dimension 8”, Ann. of Math. $$(2) 185$$:3 (2017), 991-1015. · Zbl 1373.52025  D. · Zbl 1259.11042
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