Eisenstein-series evaluations for a family of hyperbolic cosine Lambert series
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912977737744384 |
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| author | Kalinin, Nikita |
| author_facet | Kalinin, Nikita |
| contents | We study a family of hyperbolic Lambert series of the form \[ S_m=\sum_{n=1}^\infty\left( \frac{n^{2m}}{\cosh(πn)-1} -\frac{(2^{2m+1}-(-1)^{m(m+1)/2}2^{m+1}+4) n^{2m}}{\cosh(2πn)-1} +\frac{2^{2m+2}n^{2m}}{\cosh(4πn)-1} \right). \] We prove that \[ S_0=\frac1{12},\qquad S_1=\frac1{2π^2},\qquad S_m=0 \quad (m>1). \] We also evaluate the quadratic hyperbolic series \[ \sum_{n=1}^\infty \left( \frac{4}{(\cosh(πn)-1)^2} -\frac{55}{(\cosh(2πn)-1)^2} +\frac{16}{(\cosh(4πn)-1)^2} \right) = \frac{77-234/π}{72}. \] The proof is based on rewriting the hyperbolic kernels as Lambert series and identifying the resulting sums with derivatives of Eisenstein series at the CM points $i/2$, $i$, and $2i$. The initial evaluations are reduced to explicit identities for $E_2$, $E_4$, and $E_6$, together with a theta-constant relation at $2i$, while the general vanishing result is obtained from a parity decomposition of the Gaussian lattice in the classical Eisenstein series $G_k$. This gives a uniform modular explanation for a family of hyperbolic cosine identities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21265 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eisenstein-series evaluations for a family of hyperbolic cosine Lambert series Kalinin, Nikita Number Theory 11F11, 11F27, 33E05, 33C05 We study a family of hyperbolic Lambert series of the form \[ S_m=\sum_{n=1}^\infty\left( \frac{n^{2m}}{\cosh(πn)-1} -\frac{(2^{2m+1}-(-1)^{m(m+1)/2}2^{m+1}+4) n^{2m}}{\cosh(2πn)-1} +\frac{2^{2m+2}n^{2m}}{\cosh(4πn)-1} \right). \] We prove that \[ S_0=\frac1{12},\qquad S_1=\frac1{2π^2},\qquad S_m=0 \quad (m>1). \] We also evaluate the quadratic hyperbolic series \[ \sum_{n=1}^\infty \left( \frac{4}{(\cosh(πn)-1)^2} -\frac{55}{(\cosh(2πn)-1)^2} +\frac{16}{(\cosh(4πn)-1)^2} \right) = \frac{77-234/π}{72}. \] The proof is based on rewriting the hyperbolic kernels as Lambert series and identifying the resulting sums with derivatives of Eisenstein series at the CM points $i/2$, $i$, and $2i$. The initial evaluations are reduced to explicit identities for $E_2$, $E_4$, and $E_6$, together with a theta-constant relation at $2i$, while the general vanishing result is obtained from a parity decomposition of the Gaussian lattice in the classical Eisenstein series $G_k$. This gives a uniform modular explanation for a family of hyperbolic cosine identities. |
| title | Eisenstein-series evaluations for a family of hyperbolic cosine Lambert series |
| topic | Number Theory 11F11, 11F27, 33E05, 33C05 |
| url | https://arxiv.org/abs/2603.21265 |