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I am interested in the following function:

$$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$

This function is analogous to the Weierstrass $\wp$ function, the only difference being the use of absolute values under the squares. Has such a function already been referenced in the litterature?

Alternatively, would it be possible to give it a representation in terms of known elliptic functions?

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    $\begingroup$ This function is not going to be holomorphic, which very much limits how well we can study it using complex methods. $\endgroup$
    – Wojowu
    Commented Mar 6 at 17:11
  • $\begingroup$ Hmmm, I understand... Now I suspect that: $$\wp(z)*\wp(z)^*$$, could be equal to $$\sum_{w\in L^*} \frac{1}{|z-w|^4}-\frac{1}{|w|^4}$$ If only conjugate terms add up constructively and the others cancel out. Could this be the case? (that is another question I realize...) $\endgroup$
    – Aobara
    Commented Mar 6 at 17:11

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Sums of this kind are called Epstein Zeta function.

More generally, they are studied under the name of Lattice Sums. There is the book Lattice sums then and now devoted to this subject. There one can find many different representations for these sums as fast converging series or integrals (exponentially fast converging), suitable for both numerical and analytical calculations.

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  • $\begingroup$ Great reference! Thank you $\endgroup$
    – Aobara
    Commented Mar 7 at 10:47

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