Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity
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| Format: | Preprint |
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2024
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| _version_ | 1866911448799641600 |
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| author | Kalinin, Nikita |
| author_facet | Kalinin, Nikita |
| contents | We study lattice sums $\sum \frac{1}{(\|x\|\|y\|\|x+y\|)^s}$ taken over $SL_+(2,\mathbb Z)$, i.e.\ the set of pairs $(x,y)$ of primitive lattice vectors in $\mathbb Z_{\geq 0}^2$ with $\det(x, y) = 1$. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on $SL_+(2,\mathbb Z)$ that yields, in particular, $$\sum_{(x,y)\in SL_+(2,\mathbb Z)} \frac{1}{\|x\|^2\,\|y\|^2\,\|x+y\|^2}=\fracπ{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+π^3ζ(3)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity Kalinin, Nikita Number Theory 11M41, 11P21, 33B15, 11Y60, 11F03 We study lattice sums $\sum \frac{1}{(\|x\|\|y\|\|x+y\|)^s}$ taken over $SL_+(2,\mathbb Z)$, i.e.\ the set of pairs $(x,y)$ of primitive lattice vectors in $\mathbb Z_{\geq 0}^2$ with $\det(x, y) = 1$. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on $SL_+(2,\mathbb Z)$ that yields, in particular, $$\sum_{(x,y)\in SL_+(2,\mathbb Z)} \frac{1}{\|x\|^2\,\|y\|^2\,\|x+y\|^2}=\fracπ{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+π^3ζ(3)$. |
| title | Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity |
| topic | Number Theory 11M41, 11P21, 33B15, 11Y60, 11F03 |
| url | https://arxiv.org/abs/2410.10884 |