Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912642588737536 |
|---|---|
| author | Sathyanarayana, Sumukha Sharan, N. Guru |
| author_facet | Sathyanarayana, Sumukha Sharan, N. Guru |
| contents | In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $Θ(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of $Θ(r,s,t,x)$ in terms of Herglotz-Zagier type functions, and their derivatives. As applications of this, we derive Kronecker limit type formula in the second variable and a new infinite family of modular relations called mixed functional equations. We also study the zeroes, special values and singularities of the above function when all its arguments $r,s$ and $t$ are equal, which builds on a few earlier results due to Romik. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10093 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values Sathyanarayana, Sumukha Sharan, N. Guru Number Theory 11M32, 30D30 In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $Θ(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of $Θ(r,s,t,x)$ in terms of Herglotz-Zagier type functions, and their derivatives. As applications of this, we derive Kronecker limit type formula in the second variable and a new infinite family of modular relations called mixed functional equations. We also study the zeroes, special values and singularities of the above function when all its arguments $r,s$ and $t$ are equal, which builds on a few earlier results due to Romik. |
| title | Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values |
| topic | Number Theory 11M32, 30D30 |
| url | https://arxiv.org/abs/2510.10093 |