Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions
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| Format: | Preprint |
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2025
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| author | Sathyanarayana, Sumukha Sharan, N. Guru |
| author_facet | Sathyanarayana, Sumukha Sharan, N. Guru |
| contents | We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $Θ(r,r,t,x)$ as a function of the third argument around $t=1-r$. We then show that the above Kronecker limit type formula is equivalent to the two-term functional equation for the higher Herglotz function obtained by Vlasenko and Zagier. We also show the equivalence between a previously known Kronecker limit type formula for $Θ(1,1,t,x)$ around $t=0$ and the two-term functional equation for the Herglotz-Zagier function obtained by Zagier. Using the theory of the Mordell-Tornheim zeta function, we obtain results of Ramanujan, Guinand, Zagier, and Vlasenko-Zagier as consequences, to further show that the Mordell-Tornheim zeta function lies centrally between many modular relations in the literature, thus providing the means to view them under one umbrella. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions Sathyanarayana, Sumukha Sharan, N. Guru Number Theory 11M32, 30D30 We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $Θ(r,r,t,x)$ as a function of the third argument around $t=1-r$. We then show that the above Kronecker limit type formula is equivalent to the two-term functional equation for the higher Herglotz function obtained by Vlasenko and Zagier. We also show the equivalence between a previously known Kronecker limit type formula for $Θ(1,1,t,x)$ around $t=0$ and the two-term functional equation for the Herglotz-Zagier function obtained by Zagier. Using the theory of the Mordell-Tornheim zeta function, we obtain results of Ramanujan, Guinand, Zagier, and Vlasenko-Zagier as consequences, to further show that the Mordell-Tornheim zeta function lies centrally between many modular relations in the literature, thus providing the means to view them under one umbrella. |
| title | Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions |
| topic | Number Theory 11M32, 30D30 |
| url | https://arxiv.org/abs/2510.10088 |